{"id":14802,"date":"2018-09-27T18:36:23","date_gmt":"2018-09-27T18:36:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/geometric-sequences-2\/"},"modified":"2021-06-30T15:14:32","modified_gmt":"2021-06-30T15:14:32","slug":"geometric-sequences-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/chapter\/geometric-sequences-2\/","title":{"raw":"Geometric Sequences","rendered":"Geometric Sequences"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the common ratio for a geometric sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Give terms of a geometric sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Write the formula for a geometric sequence.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Finding Common Ratios<\/h3>\r\nA <strong>geometric sequence<\/strong>\u00a0changes by a constant factor. Each term of a geometric sequence increases or decreases by a constant factor called the <strong>common ratio<\/strong>. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183608\/CNX_Precalc_Figure_11_03_0012.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Definition of a Geometric Sequence<\/h3>\r\nA <strong>geometric sequence<\/strong> is one in which any term divided by the previous term is a constant. This constant is called the <strong>common ratio<\/strong> of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If [latex]{a}_{1}[\/latex] is the initial term of a geometric sequence and [latex]r[\/latex] is the common ratio, the sequence will be\r\n<div style=\"text-align: center;\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex].<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/h3>\r\n<ol>\r\n \t<li>Divide each term by the previous term.<\/li>\r\n \t<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding Common Ratios<\/h3>\r\nIs the sequence geometric? If so, find the common ratio.\r\n<ol>\r\n \t<li>[latex]1\\text{,}2\\text{,}4\\text{,}8\\text{,}16\\text{,}..[\/latex].<\/li>\r\n \t<li>[latex]48\\text{,}12\\text{,}4\\text{, }2\\text{,}..[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"507468\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507468\"]\r\n\r\nDivide each term by the previous term to determine whether a common ratio exists.\r\n<ol>\r\n \t<li>[latex]\\begin{align}&amp;\\frac{2}{1}=2&amp;&amp; \\frac{4}{2}=2&amp;&amp; \\frac{8}{4}=2&amp;&amp; \\frac{16}{8}=2 \\end{align}[\/latex]\r\nThe sequence is geometric because there is a common ratio. The common ratio is 2.<\/li>\r\n \t<li>[latex]\\begin{align}&amp;\\frac{12}{48}=\\frac{1}{4}&amp;&amp; \\frac{4}{12}=\\frac{1}{3}&amp;&amp; \\frac{2}{4}=\\frac{1}{2} \\end{align}[\/latex]\r\nThe sequence is not geometric because there is not a common ratio.<\/li>\r\n<\/ol>\r\n<div><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIs the sequence geometric? If so, find the common ratio.\r\n<p style=\"text-align: center;\">[latex]5,10,15,20,\\dots.[\/latex]<\/p>\r\n[reveal-answer q=\"231805\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231805\"]\r\n\r\nThe sequence is not geometric because [latex]\\frac{10}{5}\\ne \\frac{15}{10}[\/latex] .\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIs the sequence geometric? If so, find the common ratio.\r\n<p style=\"text-align: center;\">[latex]100,20,4,\\frac{4}{5},\\dots[\/latex]<\/p>\r\n[reveal-answer q=\"993850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"993850\"]\r\n\r\nThe sequence is geometric. The common ratio is [latex]\\frac{1}{5}[\/latex] .\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174802[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Writing Terms of Geometric Sequences<\/h2>\r\nNow 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.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{1}=-2 \\\\ &amp;{a}_{2}=\\left(-2\\cdot 4\\right)=-8 \\\\ &amp;{a}_{3}=\\left(-8\\cdot 4\\right)=-32 \\\\ &amp;{a}_{4}=\\left(-32\\cdot 4\\right)-128\\end{align}[\/latex]<\/div>\r\nThe first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\r\n<ol id=\"fs-id1165137409884\">\r\n \t<li>Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\r\n \t<li>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>\r\n \t<li>Write the terms separated by commons within brackets.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Writing the Terms of a Geometric Sequence<\/h3>\r\nList the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].\r\n\r\n[reveal-answer q=\"890639\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"890639\"]\r\n\r\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.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{1}=5 \\\\ &amp;{a}_{2}=-2{a}_{1}=-10\\\\ &amp;{a}_{3}=-2{a}_{2}=20\\\\ &amp;{a}_{4}=-2{a}_{3}=-40\\end{align}[\/latex]<\/p>\r\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].\r\n\r\n[reveal-answer q=\"779452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"779452\"]\r\n\r\n[latex]\\left\\{18,6,2,\\frac{2}{3},\\frac{2}{9}\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]172712[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using Explicit Formulas for Geometric Sequences<\/h2>\r\nWe can write explicit formulas that allow us to find particular terms.\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\r\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. An explicit formula for this sequence is\r\n<div style=\"text-align: center;\">[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Explicit Formula for a Geometric Sequence<\/h3>\r\nThe <em>n<\/em>th term of a geometric sequence is given by the <strong>explicit formula<\/strong>:\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\r\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\r\n\r\n[reveal-answer q=\"86951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"86951\"]\r\n\r\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\r\n<p style=\"text-align: center;\">[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/p>\r\nFind the common ratio using the given fourth term.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}{r}^{n - 1}\\\\ &amp;{a}_{4}=3{r}^{3}&amp;&amp; \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r \\\\ &amp;24=3{r}^{3}&amp;&amp; \\text{Substitute }24\\text{ for}{a}_{4} \\\\ &amp;8={r}^{3}&amp;&amp; \\text{Divide} \\\\ &amp;r=2&amp;&amp; \\text{Solve for the common ratio} \\end{align}[\/latex]<\/p>\r\nFind the second term by multiplying the first term by the common ratio.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{2}&amp; =2{a}_{1} \\\\ &amp;=2\\left(3\\right) \\\\ &amp;=6 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\r\n\r\n[reveal-answer q=\"947398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"947398\"]\r\n\r\n[latex]{a}_{6}=16,384[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Writing an Explicit Formula for the <em>n<\/em>th Term of a Geometric Sequence<\/h3>\r\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"862104\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862104\"]\r\n\r\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\r\n<p style=\"text-align: center;\">[latex]\\frac{10}{2}=5[\/latex]<\/p>\r\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{n}&amp;={a}_{1}{r}^{\\left(n - 1\\right)}\\\\ {a}_{n}&amp;=2\\cdot {5}^{n - 1}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite an explicit formula for the following geometric sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"367043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"367043\"]\r\n\r\n[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]172720[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Solving Application Problems with Geometric Sequences<\/h2>\r\nIn real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Solving Application Problems with Geometric Sequences<\/h3>\r\nIn 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.\r\n<ol>\r\n \t<li>Write a formula for the student population.<\/li>\r\n \t<li>Estimate the student population in 2020.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"891252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"891252\"]\r\n<ol>\r\n \t<li>The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04. Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get\r\n<div style=\"text-align: center;\">[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div><\/li>\r\n \t<li>We can find the number of years since 2013 by subtracting.\r\n<div style=\"text-align: center;\">[latex]2020 - 2013=7[\/latex]<\/div>\r\nWe are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.\r\n<div style=\"text-align: center;\">[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\r\nThe student population will be about 374 in 2020.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nA business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.\r\n\r\na. Write a formula for the number of hits.\r\n\r\nb. Estimate the number of hits in 5 weeks.\r\n\r\n[reveal-answer q=\"68120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"68120\"]\r\n\r\na.\u00a0[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]\r\nb. The number of hits will be about 333.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li style=\"font-weight: 400;\">Find the common ratio for a geometric sequence.<\/li>\n<li style=\"font-weight: 400;\">Give terms of a geometric sequence.<\/li>\n<li style=\"font-weight: 400;\">Write the formula for a geometric sequence.<\/li>\n<\/ul>\n<\/div>\n<h3>Finding Common Ratios<\/h3>\n<p>A <strong>geometric sequence<\/strong>\u00a0changes by a constant factor. Each term of a geometric sequence increases or decreases by a constant factor called the <strong>common ratio<\/strong>. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183608\/CNX_Precalc_Figure_11_03_0012.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Definition of a Geometric Sequence<\/h3>\n<p>A <strong>geometric sequence<\/strong> is one in which any term divided by the previous term is a constant. This constant is called the <strong>common ratio<\/strong> of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If [latex]{a}_{1}[\/latex] is the initial term of a geometric sequence and [latex]r[\/latex] is the common ratio, the sequence will be<\/p>\n<div style=\"text-align: center;\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex].<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/h3>\n<ol>\n<li>Divide each term by the previous term.<\/li>\n<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding Common Ratios<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<ol>\n<li>[latex]1\\text{,}2\\text{,}4\\text{,}8\\text{,}16\\text{,}..[\/latex].<\/li>\n<li>[latex]48\\text{,}12\\text{,}4\\text{, }2\\text{,}..[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q507468\">Show Solution<\/span><\/p>\n<div id=\"q507468\" class=\"hidden-answer\" style=\"display: none\">\n<p>Divide each term by the previous term to determine whether a common ratio exists.<\/p>\n<ol>\n<li>[latex]\\begin{align}&\\frac{2}{1}=2&& \\frac{4}{2}=2&& \\frac{8}{4}=2&& \\frac{16}{8}=2 \\end{align}[\/latex]<br \/>\nThe sequence is geometric because there is a common ratio. The common ratio is 2.<\/li>\n<li>[latex]\\begin{align}&\\frac{12}{48}=\\frac{1}{4}&& \\frac{4}{12}=\\frac{1}{3}&& \\frac{2}{4}=\\frac{1}{2} \\end{align}[\/latex]<br \/>\nThe sequence is not geometric because there is not a common ratio.<\/li>\n<\/ol>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<p style=\"text-align: center;\">[latex]5,10,15,20,\\dots.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q231805\">Show Solution<\/span><\/p>\n<div id=\"q231805\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is not geometric because [latex]\\frac{10}{5}\\ne \\frac{15}{10}[\/latex] .<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<p style=\"text-align: center;\">[latex]100,20,4,\\frac{4}{5},\\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993850\">Show Solution<\/span><\/p>\n<div id=\"q993850\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is geometric. The common ratio is [latex]\\frac{1}{5}[\/latex] .<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174802\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174802&theme=oea&iframe_resize_id=ohm174802\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Writing Terms of Geometric Sequences<\/h2>\n<p>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.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&{a}_{1}=-2 \\\\ &{a}_{2}=\\left(-2\\cdot 4\\right)=-8 \\\\ &{a}_{3}=\\left(-8\\cdot 4\\right)=-32 \\\\ &{a}_{4}=\\left(-32\\cdot 4\\right)-128\\end{align}[\/latex]<\/div>\n<p>The first four terms are [latex]\\left\\{-2\\text{, }-8\\text{, }-32\\text{, }-128\\right\\}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n<ol id=\"fs-id1165137409884\">\n<li>Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n<li>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<li>Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Writing the Terms of a Geometric Sequence<\/h3>\n<p>List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q890639\">Show Solution<\/span><\/p>\n<div id=\"q890639\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{1}=5 \\\\ &{a}_{2}=-2{a}_{1}=-10\\\\ &{a}_{3}=-2{a}_{2}=20\\\\ &{a}_{4}=-2{a}_{3}=-40\\end{align}[\/latex]<\/p>\n<p>The first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q779452\">Show Solution<\/span><\/p>\n<div id=\"q779452\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{18,6,2,\\frac{2}{3},\\frac{2}{9}\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm172712\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172712&theme=oea&iframe_resize_id=ohm172712\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Explicit Formulas for Geometric Sequences<\/h2>\n<p>We can write explicit formulas that allow us to find particular terms.<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<p>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. An explicit formula for this sequence is<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Explicit Formula for a Geometric Sequence<\/h3>\n<p>The <em>n<\/em>th term of a geometric sequence is given by the <strong>explicit formula<\/strong>:<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n<p>Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q86951\">Show Solution<\/span><\/p>\n<div id=\"q86951\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3,3r,3{r}^{2},3{r}^{3},..[\/latex].<\/p>\n<p>Find the common ratio using the given fourth term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{1}{r}^{n - 1}\\\\ &{a}_{4}=3{r}^{3}&& \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\text{and }r \\\\ &24=3{r}^{3}&& \\text{Substitute }24\\text{ for}{a}_{4} \\\\ &8={r}^{3}&& \\text{Divide} \\\\ &r=2&& \\text{Solve for the common ratio} \\end{align}[\/latex]<\/p>\n<p>Find the second term by multiplying the first term by the common ratio.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{2}& =2{a}_{1} \\\\ &=2\\left(3\\right) \\\\ &=6 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q947398\">Show Solution<\/span><\/p>\n<div id=\"q947398\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{6}=16,384[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Writing an Explicit Formula for the <em>n<\/em>th Term of a Geometric Sequence<\/h3>\n<p>Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862104\">Show Solution<\/span><\/p>\n<div id=\"q862104\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{10}{2}=5[\/latex]<\/p>\n<p>The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{n}&={a}_{1}{r}^{\\left(n - 1\\right)}\\\\ {a}_{n}&=2\\cdot {5}^{n - 1}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Write an explicit formula for the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{-1\\text{, }3\\text{, }-9\\text{, }27\\text{, }\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367043\">Show Solution<\/span><\/p>\n<div id=\"q367043\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm172720\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172720&theme=oea&iframe_resize_id=ohm172720\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Solving Application Problems with Geometric Sequences<\/h2>\n<p>In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Solving Application Problems with Geometric Sequences<\/h3>\n<p>In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.<\/p>\n<ol>\n<li>Write a formula for the student population.<\/li>\n<li>Estimate the student population in 2020.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q891252\">Show Solution<\/span><\/p>\n<div id=\"q891252\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04. Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get\n<div style=\"text-align: center;\">[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div>\n<\/li>\n<li>We can find the number of years since 2013 by subtracting.\n<div style=\"text-align: center;\">[latex]2020 - 2013=7[\/latex]<\/div>\n<p>We are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.<\/p>\n<div style=\"text-align: center;\">[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\n<p>The student population will be about 374 in 2020.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.<\/p>\n<p>a. Write a formula for the number of hits.<\/p>\n<p>b. Estimate the number of hits in 5 weeks.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q68120\">Show Solution<\/span><\/p>\n<div id=\"q68120\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a0[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]<br \/>\nb. The number of hits will be about 333.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-14802\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">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":17533,"menu_order":6,"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-14802","chapter","type-chapter","status-publish","hentry"],"part":14758,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14802","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14802\/revisions"}],"predecessor-version":[{"id":16148,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14802\/revisions\/16148"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/parts\/14758"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14802\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/media?parent=14802"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapter-type?post=14802"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/contributor?post=14802"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/license?post=14802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}