How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n\t
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n\t
- Write the terms separated by commons within brackets.<\/li>\n<\/ol><\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\nList the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nMultiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.\n[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\n\n<\/div>\n\nTry It 3<\/h3>\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].\n\nSolution<\/a>\n\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\nA recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is\n[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n- State the initial term.<\/li>\n\t
- Find the common ratio by dividing any term by the preceding term.<\/li>\n\t
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol><\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202541\/CNX_Precalc_Figure_11_03_0032.jpg\")
Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202542\/CNX_Precalc_Figure_11_03_0042.jpg\")
Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202544\/CNX_Precalc_Figure_11_03_0052.jpg\")
Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\nList the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nMultiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.\n[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\n\n<\/div>\n\nTry It 3<\/h3>\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].\n\nSolution<\/a>\n\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\nA recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is\n[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n- State the initial term.<\/li>\n\t
- Find the common ratio by dividing any term by the preceding term.<\/li>\n\t
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol><\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Solution<\/h3>\nMultiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.\n[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\n\n<\/div>\n\nTry It 3<\/h3>\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].\n\nSolution<\/a>\n\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\nA recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is\n[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n- State the initial term.<\/li>\n\t
- Find the common ratio by dividing any term by the preceding term.<\/li>\n\t
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol><\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Try It 3<\/h3>\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].\n\nSolution<\/a>\n\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\nA recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is\n[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n- State the initial term.<\/li>\n\t
- Find the common ratio by dividing any term by the preceding term.<\/li>\n\t
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol><\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
A General Note: Recursive Formula for a Geometric Sequence<\/h3>\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is\n[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n- State the initial term.<\/li>\n\t
- Find the common ratio by dividing any term by the preceding term.<\/li>\n\t
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol><\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
How To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n- State the initial term.<\/li>\n\t
- Find the common ratio by dividing any term by the preceding term.<\/li>\n\t
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol><\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Solution<\/h3>\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\n[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\n[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Analysis of the Solution<\/h3>\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 2<\/b>[\/caption]\n\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Q & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em>\n\n<\/div>\n\nTry It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Try It 4<\/h3>\nWrite a recursive formula for the following geometric sequence.\n[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n\nSolution<\/a>\n\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Using Explicit Formulas for Geometric Sequences<\/h2>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 3<\/b>[\/caption]\n\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Figure 3<\/b>[\/caption]\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Example 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n<\/div>\n\nSolution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Solution<\/h3>\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.\n[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.\n[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Analysis of the Solution<\/h3>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n<\/div>\n\nTry It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Try It 5<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\nSolution<\/a>\n\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Example 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Solution<\/h3>\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]Figure 4<\/b>[\/caption]\n\n<\/div>\n\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Figure 4<\/b>[\/caption]\n\n<\/div>\nTry It 6<\/h3>\nWrite an explicit formula for the following geometric sequence.\n[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>","rendered":"Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)-128\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
\nHow To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<\/p>\n
The first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n
How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n\n- Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
- Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
- Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n\n
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
\nTry It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\nA General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n
[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/div>\n<\/div>\n\nHow To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n\n- State the initial term.<\/li>\n
- Find the common ratio by dividing any term by the preceding term.<\/li>\n
- Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n
[latex]\\begin{array}{l}{a}_{n}=r{a}_{n - 1}\\\\ {a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n
<\/p>\nFigure 2<\/b><\/p>\n<\/div>\n<\/div>\n\nQ & A<\/h3>\nDo we have to divide the second term by the first term to find the common ratio?<\/h3>\n
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n\nTry It 4<\/h3>\n
Write a recursive formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/p>\n<\/div>\nUsing Explicit Formulas for Geometric Sequences<\/h2>\n
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n
[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n
[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\nThe graph of the sequence is shown in Figure 3.<\/p>\n
<\/p>\nFigure 3<\/b><\/p>\n<\/div>\n\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n\nExample 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n
\nSolution<\/h3>\n
The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n
[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/div>\nFind the common ratio using the given fourth term.<\/p>\n
[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n - 1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}{a}_{4}\\hfill \\\\ 8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\nFind the second term by multiplying the first term by the common ratio.<\/p>\n
[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n\nAnalysis of the Solution<\/h3>\n
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n
\nTry It 5<\/h3>\n
Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n\nExample 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n
[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n\nSolution<\/h3>\n
The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n
[latex]\\frac{10}{2}=5[\/latex]<\/div>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n
[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n - 1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n
<\/p>\nFigure 4<\/b><\/p>\n<\/div>\n<\/div>\n\nTry It 6<\/h3>\n
Write an explicit formula for the following geometric sequence.<\/p>\n
[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\nSolution<\/a><\/div>\n<\/div>\n\n\t\t\t \n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>- Precalculus. Authored by<\/strong>: OpenStax College. Provided by<\/strong>: OpenStax. Located at<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. License<\/strong>: CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":2049,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":2168,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/2168"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2049"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<\/div>\n
Solution<\/h3>\n
Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.<\/p>\n
The first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n
Try It 3<\/h3>\n
List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n
Solution<\/a><\/p>\n<\/div>\n A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n Write a recursive formula for the following geometric sequence.<\/p>\n The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n Substitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.<\/p>\n Figure 2<\/b><\/p>\n<\/div>\n<\/div>\n No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n Write a recursive formula for the following geometric sequence.<\/p>\n Solution<\/a><\/p>\n<\/div>\n Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n Let\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n The graph of the sequence is shown in Figure 3.<\/p>\n Figure 3<\/b><\/p>\n<\/div>\n The n<\/em>th term of a geometric sequence is given by the explicit formula<\/strong>:<\/p>\n Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<\/div>\n The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n Find the common ratio using the given fourth term.<\/p>\n Find the second term by multiplying the first term by the common ratio.<\/p>\n The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n Solution<\/a><\/p>\n<\/div>\n Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n The graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n Figure 4<\/b><\/p>\n<\/div>\n<\/div>\n Write an explicit formula for the following geometric sequence.<\/p>\nUsing Recursive Formulas for Geometric Sequences<\/h2>\n
A General Note: Recursive Formula for a Geometric Sequence<\/h3>\n
How To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n
\n
Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n
Solution<\/h3>\n
Analysis of the Solution<\/h3>\n
<\/p>\nQ & A<\/h3>\n
Do we have to divide the second term by the first term to find the common ratio?<\/h3>\n
Try It 4<\/h3>\n
Using Explicit Formulas for Geometric Sequences<\/h2>\n
<\/p>\nA General Note: Explicit Formula for a Geometric Sequence<\/h3>\n
Example 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n
Solution<\/h3>\n
Analysis of the Solution<\/h3>\n
Try It 5<\/h3>\n
Example 6: Writing an Explicit Formula for the n<\/em>th Term of a Geometric Sequence<\/h3>\n
Solution<\/h3>\n
<\/p>\nTry It 6<\/h3>\n
Candela Citations<\/h3>\n\t\t\t\t\t