{"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":"2019-09-09T21:30:38","modified_gmt":"2019-09-09T21:30:38","slug":"geometric-sequences-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/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\nThe yearly salary values described form a <strong>geometric sequence<\/strong> because they change by a constant factor each year. 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>\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183611\/CNX_Precalc_Figure_11_03_0022.jpg\" alt=\"Graph of two sequences where graph (a) is geometric and graph (b) is exponential.\" width=\"975\" height=\"286\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?<\/h3>\r\n<em>No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.<\/em>\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 Recursive Formulas for Geometric Sequences<\/h2>\r\nA <strong>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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Recursive Formula for a Geometric Sequence<\/h3>\r\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is\r\n<p style=\"text-align: center;\">[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\r\n<ol id=\"fs-id1165137442323\">\r\n \t<li>State the initial term.<\/li>\r\n \t<li>Find the common ratio by dividing any term by the preceding term.<\/li>\r\n \t<li>Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\r\nWrite a recursive formula for the following geometric sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"427299\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"427299\"]\r\n\r\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\r\n<p style=\"text-align: center;\">[latex]r=\\frac{9}{6}=1.5[\/latex]<\/p>\r\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}=r{a}_{n - 1}\\\\ &amp;{a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ &amp;{a}_{1}=6\\end{align}[\/latex]<\/p>\r\n\r\n<h3>Analysis of the Solution<\/h3>\r\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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183613\/CNX_Precalc_Figure_11_03_0032.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"215\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>Do we have to divide the second term by the first term to find the common ratio?<\/h3>\r\n<em>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>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite a recursive formula for the following geometric sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"625241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625241\"]\r\n\r\n[latex]\\begin{align}&amp;{a}_{1}=2\\\\ &amp;{a}_{n}=\\frac{2}{3}{a}_{n - 1}\\text{ for }n\\ge 2\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Using Explicit Formulas for Geometric Sequences<\/h2>\r\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.\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 and an exponential function with a base 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\nThe graph of the sequence is shown in Figure 3.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183615\/CNX_Precalc_Figure_11_03_0042.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"440\" \/> <b>Figure 3<\/b>[\/caption]\r\n\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\r\n<div>\r\n<h4>Analysis of the Solution<\/h4>\r\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.\r\n\r\n<\/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\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\nThe graph of this sequence in Figure 4\u00a0shows an exponential pattern.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183617\/CNX_Precalc_Figure_11_03_0052.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"290\" \/> <b>Figure 4<\/b>[\/caption]\r\n\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>\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165133155748\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\r\n<td>[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>explicit formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\r\n<td>[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\r\n \t<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\r\n \t<li>The common ratio can be found by dividing any term in the sequence by the previous term.<\/li>\r\n \t<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/li>\r\n \t<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[\/latex] for [latex]n\\ge 2[\/latex] .<\/li>\r\n \t<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\r\n \t<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex].<\/li>\r\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137740810\" class=\"definition\">\r\n \t<dt>common ratio<\/dt>\r\n \t<dd id=\"fs-id1165137849293\">the ratio between any two consecutive terms in a geometric sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137611024\" class=\"definition\">\r\n \t<dt>geometric sequence<\/dt>\r\n \t<dd id=\"fs-id1165137673421\">a sequence in which the ratio of a term to a previous term is a constant<\/dd>\r\n<\/dl>","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>The yearly salary values described form a <strong>geometric sequence<\/strong> because they change by a constant factor each year. 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>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183611\/CNX_Precalc_Figure_11_03_0022.jpg\" alt=\"Graph of two sequences where graph (a) is geometric and graph (b) is exponential.\" width=\"975\" height=\"286\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?<\/h3>\n<p><em>No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.<\/em><\/p>\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 Recursive Formulas for Geometric Sequences<\/h2>\n<p>A <strong>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<div class=\"textbox\">\n<h3>A General Note: Recursive Formula for a Geometric Sequence<\/h3>\n<p>The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first several terms of a geometric sequence, write its recursive formula.<\/h3>\n<ol id=\"fs-id1165137442323\">\n<li>State the initial term.<\/li>\n<li>Find the common ratio by dividing any term by the preceding term.<\/li>\n<li>Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Using Recursive Formulas for Geometric Sequences<\/h3>\n<p>Write a recursive formula for the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q427299\">Show Solution<\/span><\/p>\n<div id=\"q427299\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n<p style=\"text-align: center;\">[latex]r=\\frac{9}{6}=1.5[\/latex]<\/p>\n<p>Substitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}=r{a}_{n - 1}\\\\ &{a}_{n}=1.5{a}_{n - 1}\\text{ for }n\\ge 2\\\\ &{a}_{1}=6\\end{align}[\/latex]<\/p>\n<h3>Analysis of the Solution<\/h3>\n<p>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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183613\/CNX_Precalc_Figure_11_03_0032.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"215\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Do we have to divide the second term by the first term to find the common ratio?<\/h3>\n<p><em>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<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Write a recursive formula for the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q625241\">Show Solution<\/span><\/p>\n<div id=\"q625241\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}&{a}_{1}=2\\\\ &{a}_{n}=\\frac{2}{3}{a}_{n - 1}\\text{ for }n\\ge 2\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Using Explicit Formulas for Geometric Sequences<\/h2>\n<p>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<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 and an exponential function with a base 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<p>The graph of the sequence is shown in Figure 3.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183615\/CNX_Precalc_Figure_11_03_0042.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/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<h4>Analysis of the Solution<\/h4>\n<p>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<\/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<p>The graph of this sequence in Figure 4\u00a0shows an exponential pattern.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183617\/CNX_Precalc_Figure_11_03_0052.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"290\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\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<h2>Key Equations<\/h2>\n<table id=\"eip-id1165133155748\" summary=\"..\">\n<tbody>\n<tr>\n<td>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\n<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\n<li>The common ratio can be found by dividing any term in the sequence by the previous term.<\/li>\n<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/li>\n<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[\/latex] for [latex]n\\ge 2[\/latex] .<\/li>\n<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex].<\/li>\n<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137740810\" class=\"definition\">\n<dt>common ratio<\/dt>\n<dd id=\"fs-id1165137849293\">the ratio between any two consecutive terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137611024\" class=\"definition\">\n<dt>geometric sequence<\/dt>\n<dd id=\"fs-id1165137673421\">a sequence in which the ratio of a term to a previous term is a constant<\/dd>\n<\/dl>\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":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-14802","chapter","type-chapter","status-publish","hentry"],"part":14758,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14802","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14802\/revisions"}],"predecessor-version":[{"id":15857,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14802\/revisions\/15857"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14758"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14802\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=14802"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=14802"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=14802"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=14802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}