{"id":441,"date":"2019-07-15T22:45:04","date_gmt":"2019-07-15T22:45:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/finding-common-differences\/"},"modified":"2019-07-15T22:45:04","modified_gmt":"2019-07-15T22:45:04","slug":"finding-common-differences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/finding-common-differences\/","title":{"raw":"Terms of an Arithmetic Sequence","rendered":"Terms of an Arithmetic Sequence"},"content":{"raw":"\n<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Find the common difference for an arithmetic sequence<\/li><li>Find terms of an arithmetic sequence<\/li><\/ul><\/div>\nThe values of the truck in the example on the previous page form an <strong>arithmetic sequence<\/strong> because they change by a constant amount each year. Each term increases or decreases by the same constant value called the <strong>common difference<\/strong> of the sequence. For this sequence the common difference is \u20133,400.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222135\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\">\n\nThe sequence below is another example of an arithmetic sequence. In this case the constant difference is 3. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add 3 to find 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\/03222137\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\">\n\n<div class=\"textbox\"><h3>A General Note: Arithmetic Sequence<\/h3>An <strong>arithmetic sequence<\/strong> is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the <strong>common difference<\/strong>. If [latex]{a}_{1}[\/latex] is the first term of an arithmetic sequence and [latex]d[\/latex] is the common difference, the sequence will be:\n[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]\n\n<\/div><div class=\"textbox exercises\"><h3>Example: Finding Common Differences<\/h3>Is each sequence arithmetic? If so, find the common difference.\n\n<ol><li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li><li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li><\/ol>[reveal-answer q=\"717238\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"717238\"]\n\nSubtract each term from the subsequent term to determine whether a common difference exists.\n\n<ol><li>The sequence is not arithmetic because there is no common difference.<div style=\"text-align: center\">[latex]\\begin{align}&amp;2-1=1 &amp;&amp; 4-2=2 &amp;&amp; 8-4=4 &amp;&amp; 16-8=8 \\end{align}[\/latex]<\/div><\/li><li>The sequence is arithmetic because there is a common difference. The common difference is 4.<div style=\"text-align: center\">[latex]\\begin{align}&amp;1-(-3)=4 &amp;&amp; 5-1=4 &amp;&amp; 9-5=4 &amp;&amp; 13-9=4 \\end{align}[\/latex]<\/div><\/li><\/ol><h4>Analysis of the Solution<\/h4>The graph of each of these sequences is shown in Figure 1. We can see from the graphs that, although both sequences show growth, [latex]a[\/latex] is not linear whereas [latex]b[\/latex] is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222143\/CNX_Precalc_Figure_11_02_0032.jpg\" alt=\"Two graphs of arithmetic sequences. Graph (a) grows exponentially while graph (b) grows linearly.\" width=\"975\" height=\"304\">\n\n[\/hidden-answer]\n\n<\/div><div class=\"textbox\"><h3>Q &amp; A<\/h3><h4>If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?<\/h4><em> No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.<\/em>\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Is the given sequence arithmetic? If so, find the common difference.\n\n<p style=\"text-align: center\">[latex]\\left\\{18,16,14,12,10,\\dots \\right\\}[\/latex]\n\n[reveal-answer q=\"463836\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"463836\"]\n\nThe sequence is arithmetic. The common difference is [latex]-2[\/latex].\n\n[\/hidden-answer]\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23735&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n\n\n<\/p><\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Is the given sequence arithmetic? If so, find the common difference.\n[latex]\\left\\{1,3,6,10,15,\\dots \\right\\}[\/latex]\n\n[reveal-answer q=\"598102\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"598102\"]\n\nThe sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].\n\n[\/hidden-answer]\n\n<\/div><h2>Writing Terms of Arithmetic Sequences<\/h2>Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[\/latex] and [latex]d[\/latex] into formula below.\n\n<p style=\"text-align: center\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]\n\n<\/p><div class=\"textbox examples\"><h3>tip for success<\/h3>The formula given above is a handy tool for calculating any term of an arithmetic sequence. As you work to memorize it, do work out the terms in the examples below individually as well to build you intuition for why the formula works.\n\n<\/div><div class=\"textbox\"><h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3><ol><li>Add the common difference to the first term to find the second term.<\/li><li>Add the common difference to the second term to find the third term.<\/li><li>Continue until all of the desired terms are identified.<\/li><li>Write the terms separated by commas within brackets.<\/li><\/ol><\/div><div class=\"textbox exercises\"><h3>Example: Writing Terms of Arithmetic Sequences<\/h3>Write the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].\n\n[reveal-answer q=\"654025\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"654025\"]\n\nAdding [latex]-3[\/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.\n\nThe first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]\n\n<h4>Analysis of the Solution<\/h4>As expected, the graph of the sequence consists of points on a line.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222146\/CNX_Precalc_Figure_11_02_0042.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"487\" height=\"250\">\n\n[\/hidden-answer]\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>List the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .\n\n[reveal-answer q=\"880961\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"880961\"]\n\n[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]\n\n[\/hidden-answer]\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5832&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n\n\n<\/div><div class=\"textbox\"><h3>How To: Given any the first term and any other term in an arithmetic sequence, find a given term.<\/h3><ol><li>Substitute the values given for [latex]{a}_{1},{a}_{n},n[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex] to solve for [latex]d[\/latex].<\/li><li>Find a given term by substituting the appropriate values for [latex]{a}_{1},n[\/latex], and [latex]d[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/li><\/ol><\/div><div class=\"textbox exercises\"><h3>Example: Writing Terms of Arithmetic Sequences<\/h3>Given [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex] .\n\n[reveal-answer q=\"644479\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"644479\"]\n\nThe sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex] .\n\n<p style=\"text-align: center\">[latex]\\left\\{8,8+d,8+2d,8+3d\\right\\}[\/latex]\n\nWe know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex] .\n\nWe can find the common difference [latex]d[\/latex] .\n\n<\/p><p style=\"text-align: center\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}+\\left(n - 1\\right)d \\\\ &amp;{a}_{4}={a}_{1}+3d \\\\ &amp;{a}_{4}=8+3d &amp;&amp; \\text{Write the fourth term of the sequence in terms of } {a}_{1} \\text{ and } d. \\\\ &amp;14=8+3d &amp;&amp; \\text{Substitute } 14 \\text{ for } {a}_{4}. \\\\ &amp;d=2 &amp;&amp; \\text{Solve for the common difference}. \\end{align}[\/latex]\n\nFind the fifth term by adding the common difference to the fourth term.\n\n<\/p><p style=\"text-align: center\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]\n\n<\/p><h4>Analysis of the Solution<\/h4>Notice that the common difference is added to 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 adding the common difference to the first term nine times or by using the equation [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].\n\n[\/hidden-answer]\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Given [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/latex] , find [latex]{a}_{2}[\/latex] .\n\n[reveal-answer q=\"20007\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"20007\"]\n\n[latex]{a}_{2}=2[\/latex]\n\n[\/hidden-answer]\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5847&amp;theme=oea&amp;iframe_resize_id=mom2[\/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 difference for an arithmetic sequence<\/li>\n<li>Find terms of an arithmetic sequence<\/li>\n<\/ul>\n<\/div>\n<p>The values of the truck in the example on the previous page form an <strong>arithmetic sequence<\/strong> because they change by a constant amount each year. Each term increases or decreases by the same constant value called the <strong>common difference<\/strong> of the sequence. For this sequence the common difference is \u20133,400.<\/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\/03222135\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" \/><\/p>\n<p>The sequence below is another example of an arithmetic sequence. In this case the constant difference is 3. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add 3 to find 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\/03222137\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Arithmetic Sequence<\/h3>\n<p>An <strong>arithmetic sequence<\/strong> is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the <strong>common difference<\/strong>. If [latex]{a}_{1}[\/latex] is the first term of an arithmetic sequence and [latex]d[\/latex] is the common difference, the sequence will be:<br \/>\n[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Common Differences<\/h3>\n<p>Is each sequence arithmetic? If so, find the common difference.<\/p>\n<ol>\n<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\n<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q717238\">Show Solution<\/span><\/p>\n<div id=\"q717238\" class=\"hidden-answer\" style=\"display: none\">\n<p>Subtract each term from the subsequent term to determine whether a common difference exists.<\/p>\n<ol>\n<li>The sequence is not arithmetic because there is no common difference.\n<div style=\"text-align: center\">[latex]\\begin{align}&2-1=1 && 4-2=2 && 8-4=4 && 16-8=8 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>The sequence is arithmetic because there is a common difference. The common difference is 4.\n<div style=\"text-align: center\">[latex]\\begin{align}&1-(-3)=4 && 5-1=4 && 9-5=4 && 13-9=4 \\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of each of these sequences is shown in Figure 1. We can see from the graphs that, although both sequences show growth, [latex]a[\/latex] is not linear whereas [latex]b[\/latex] is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.<\/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\/03222143\/CNX_Precalc_Figure_11_02_0032.jpg\" alt=\"Two graphs of arithmetic sequences. Graph (a) grows exponentially while graph (b) grows linearly.\" width=\"975\" height=\"304\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?<\/h4>\n<p><em> No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the given sequence arithmetic? If so, find the common difference.<\/p>\n<p style=\"text-align: center\">[latex]\\left\\{18,16,14,12,10,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q463836\">Show Solution<\/span><\/p>\n<div id=\"q463836\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is arithmetic. The common difference is [latex]-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm23735\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23735&#38;theme=oea&#38;iframe_resize_id=ohm23735&#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 given sequence arithmetic? If so, find the common difference.<br \/>\n[latex]\\left\\{1,3,6,10,15,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q598102\">Show Solution<\/span><\/p>\n<div id=\"q598102\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Writing Terms of Arithmetic Sequences<\/h2>\n<p>Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[\/latex] and [latex]d[\/latex] into formula below.<\/p>\n<p style=\"text-align: center\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>The formula given above is a handy tool for calculating any term of an arithmetic sequence. As you work to memorize it, do work out the terms in the examples below individually as well to build you intuition for why the formula works.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3>\n<ol>\n<li>Add the common difference to the first term to find the second term.<\/li>\n<li>Add the common difference to the second term to find the third term.<\/li>\n<li>Continue until all of the desired terms are identified.<\/li>\n<li>Write the terms separated by commas within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Arithmetic Sequences<\/h3>\n<p>Write the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654025\">Show Solution<\/span><\/p>\n<div id=\"q654025\" class=\"hidden-answer\" style=\"display: none\">\n<p>Adding [latex]-3[\/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.<\/p>\n<p>The first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>As expected, the graph of the sequence consists of points on a line.<\/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\/03222146\/CNX_Precalc_Figure_11_02_0042.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"487\" height=\"250\" \/><\/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 arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880961\">Show Solution<\/span><\/p>\n<div id=\"q880961\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm5832\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5832&#38;theme=oea&#38;iframe_resize_id=ohm5832&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given any the first term and any other term in an arithmetic sequence, find a given term.<\/h3>\n<ol>\n<li>Substitute the values given for [latex]{a}_{1},{a}_{n},n[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex] to solve for [latex]d[\/latex].<\/li>\n<li>Find a given term by substituting the appropriate values for [latex]{a}_{1},n[\/latex], and [latex]d[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Arithmetic Sequences<\/h3>\n<p>Given [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q644479\">Show Solution<\/span><\/p>\n<div id=\"q644479\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex] .<\/p>\n<p style=\"text-align: center\">[latex]\\left\\{8,8+d,8+2d,8+3d\\right\\}[\/latex]<\/p>\n<p>We know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex] .<\/p>\n<p>We can find the common difference [latex]d[\/latex] .<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{a}_{n}={a}_{1}+\\left(n - 1\\right)d \\\\ &{a}_{4}={a}_{1}+3d \\\\ &{a}_{4}=8+3d && \\text{Write the fourth term of the sequence in terms of } {a}_{1} \\text{ and } d. \\\\ &14=8+3d && \\text{Substitute } 14 \\text{ for } {a}_{4}. \\\\ &d=2 && \\text{Solve for the common difference}. \\end{align}[\/latex]<\/p>\n<p>Find the fifth term by adding the common difference to the fourth term.<\/p>\n<p style=\"text-align: center\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that the common difference is added to 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 adding the common difference to the first term nine times or by using the equation [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/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=\"q20007\">Show Solution<\/span><\/p>\n<div id=\"q20007\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{2}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm5847\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5847&#38;theme=oea&#38;iframe_resize_id=ohm5847&#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-441\">\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 5847, 5832. <strong>Authored by<\/strong>: Web-Work 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><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>Question ID 23735. <strong>Authored by<\/strong>: Roy Shahbazian. <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":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 5847, 5832\",\"author\":\"Web-Work Rochester\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 23735\",\"author\":\"Roy 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