{"id":445,"date":"2019-07-15T22:45:06","date_gmt":"2019-07-15T22:45:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/terms-of-geometric-sequences\/"},"modified":"2019-07-15T22:45:06","modified_gmt":"2019-07-15T22:45:06","slug":"terms-of-geometric-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/terms-of-geometric-sequences\/","title":{"raw":"Terms of Geometric Sequences","rendered":"Terms of Geometric Sequences"},"content":{"raw":"\n<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Find the common ratio of a geometric sequence<\/li><li>List terms of a geometric sequence given the first term and the common ratio<\/li><\/ul><\/div>\n\n<h2>Finding Common Ratios<\/h2>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.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223627\/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.\">\n\n<div class=\"textbox\"><h3>A General Note: Definition of a Geometric Sequence<\/h3>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\n\n<p style=\"text-align: center\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex].\n\n<\/p><\/div><div class=\"textbox\"><h3>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/h3><ol><li>Divide each term by the previous term.<\/li><li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li><\/ol><\/div><div class=\"textbox exercises\"><h3>Example: Finding Common Ratios<\/h3>Is the sequence geometric? If so, find the common ratio.\n\n<ol><li>[latex]1,2,4,8,16,\\dots[\/latex]<\/li><li>[latex]48,12,4,2,\\dots[\/latex]<\/li><\/ol>[reveal-answer q=\"706624\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"706624\"]\n\nDivide each term by the previous term to determine whether a common ratio exists.\n\n<ol><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]\nThe sequence is geometric because there is a common ratio. The common ratio is 2.<\/li><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]\nThe sequence is not geometric because there is not a common ratio.<\/li><\/ol><h4>Analysis of the Solution<\/h4>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.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223630\/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\">[\/hidden-answer]\n\n<\/div><div class=\"textbox\"><h3>Q &amp; A<\/h3><h4>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?<\/h4><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>\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Is the sequence geometric? If so, find the common ratio.\n\n[latex]5,10,15,20,\\dots[\/latex]\n\n[reveal-answer q=\"893960\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"893960\"]\n\nThe sequence is not geometric because [latex]\\dfrac{10}{5}\\ne \\dfrac{15}{10}[\/latex] .\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=68722&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Is the sequence geometric? If so, find the common ratio.\n\n[latex]100,20,4,\\dfrac{4}{5},\\dots[\/latex]\n\n[reveal-answer q=\"870692\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"870692\"]\n\nThe sequence is geometric. The common ratio is [latex]\\dfrac{1}{5}[\/latex] .\n\n[\/hidden-answer]\n\n<\/div><h2>Writing Terms of Geometric Sequences<\/h2>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\n<p 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]\n\nThe first four terms are [latex]\\left\\{-2,-8,-32,-128\\right\\}[\/latex].\n\n<\/p><div class=\"textbox\"><h3>How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3><ol><li>Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li><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><li>Write the terms separated by commons within brackets.<\/li><\/ol><\/div><div class=\"textbox examples\"><h3>tip for success<\/h3>You can also use the definition of a geometric sequence, given at the beginning of the section, to write out several terms.\n\n<p style=\"text-align: center\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex]\n\nTry it on the example and practice problems below.\n\n<\/p><\/div><div class=\"textbox exercises\"><h3>Example: Writing the Terms of a Geometric Sequence<\/h3>List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].\n\n[reveal-answer q=\"650557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"650557\"]\n\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],&nbsp;and so on.\n\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]\n\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\n\n[\/hidden-answer]\n\n<\/p><\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\dfrac{1}{3}[\/latex].\n\n[reveal-answer q=\"228021\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"228021\"]\n\n[latex]\\left\\{18,6,2,\\dfrac{2}{3},\\dfrac{2}{9}\\right\\}[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156689&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\n\n\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the common ratio of a geometric sequence<\/li>\n<li>List terms of a geometric sequence given the first term and the common ratio<\/li>\n<\/ul>\n<\/div>\n<h2>Finding Common Ratios<\/h2>\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\/896\/2016\/11\/03223627\/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<p style=\"text-align: center\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex].<\/p>\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 exercises\">\n<h3>Example: Finding Common Ratios<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<ol>\n<li>[latex]1,2,4,8,16,\\dots[\/latex]<\/li>\n<li>[latex]48,12,4,2,\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706624\">Show Solution<\/span><\/p>\n<div id=\"q706624\" 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<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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223630\/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\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>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?<\/h4>\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=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<p>[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=\"q893960\">Show Solution<\/span><\/p>\n<div id=\"q893960\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is not geometric because [latex]\\dfrac{10}{5}\\ne \\dfrac{15}{10}[\/latex] .<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm68722\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=68722&#38;theme=oea&#38;iframe_resize_id=ohm68722&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<p>[latex]100,20,4,\\dfrac{4}{5},\\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q870692\">Show Solution<\/span><\/p>\n<div id=\"q870692\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is geometric. The common ratio is [latex]\\dfrac{1}{5}[\/latex] .<\/p>\n<\/div>\n<\/div>\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<p 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]<\/p>\n<p>The first four terms are [latex]\\left\\{-2,-8,-32,-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>\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 examples\">\n<h3>tip for success<\/h3>\n<p>You can also use the definition of a geometric sequence, given at the beginning of the section, to write out several terms.<\/p>\n<p style=\"text-align: center\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex]<\/p>\n<p>Try it on the example and practice problems below.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: 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=\"q650557\">Show Solution<\/span><\/p>\n<div id=\"q650557\" 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],&nbsp;and 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=\"textbox key-takeaways\">\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=\\dfrac{1}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q228021\">Show Solution<\/span><\/p>\n<div id=\"q228021\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{18,6,2,\\dfrac{2}{3},\\dfrac{2}{9}\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm156689\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156689&#38;theme=oea&#38;iframe_resize_id=ohm156689&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-445\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 68722. <strong>Authored by<\/strong>: Shahbazian,Roy. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Ex: Determine if a Sequence is Arithmetic or Geometric (geometric). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Z9OwCMeohnE\">https:\/\/youtu.be\/Z9OwCMeohnE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 5856. <strong>Authored by<\/strong>: WebWork-Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 68722\",\"author\":\"Shahbazian,Roy\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and 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