{"id":2695,"date":"2024-08-12T20:41:51","date_gmt":"2024-08-12T20:41:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2695"},"modified":"2025-08-15T17:12:07","modified_gmt":"2025-08-15T17:12:07","slug":"arithmetic-sequences-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/arithmetic-sequences-learn-it-1\/","title":{"raw":"Arithmetic Sequences: Learn It 1","rendered":"Arithmetic Sequences: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the regular interval between terms in a simple sequence and use it to write the sequence's terms<\/li>\r\n \t<li>Use recursive and explicit formulas to represent and analyze arithmetic sequences<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Terms of an Arithmetic Sequence<\/h2>\r\nCompanies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">As an example, consider a woman who starts a small contracting business. She purchases a new truck for [latex]$25,000[\/latex]. After five years she estimates that she will be able to sell the truck for [latex]$8,000[\/latex].\r\n[latex]\\\\[\/latex]\r\nThe loss in value of the truck will therefore be [latex]$17,000[\/latex], which is [latex]$3,400[\/latex] per year for five years.\r\n[latex]\\\\[\/latex]\r\nThe truck will be worth [latex]$21,600[\/latex] after the first year; [latex]$18,200[\/latex] after two years; [latex]$14,800[\/latex] after three years; [latex]$11,400[\/latex] after four years; and [latex]$8,000[\/latex] at the end of five years.<\/section>The values of the truck in the example 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 [latex]\u20133,400[\/latex]. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and subtract [latex]3,400[\/latex] to find the subsequent term.\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\/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.\" width=\"487\" height=\"68\" \/> Arithmetic sequence[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>arithmetic sequence<\/h3>\r\n<div class=\"page\" title=\"Page 1074\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nAn <strong>arithmetic sequence<\/strong> is a sequence where the difference between consecutive terms is always the same.\r\n\r\n<\/div>\r\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\r\n\r\n<div class=\"column\">\r\n\r\nThe difference between consecutive terms, [latex]d[\/latex], and is called the <strong>common difference<\/strong>, for [latex]n[\/latex] greater than or equal to two.\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The sequence below is another example of an arithmetic sequence. In this case the constant difference is [latex]3[\/latex]. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add [latex]3[\/latex] to find the subsequent term.\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\/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.\" width=\"487\" height=\"68\" \/> Arithmetic sequence[\/caption]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Is each sequence arithmetic? If so, find the common difference.\r\n<ol>\r\n \t<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\r\n \t<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"717238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"717238\"]\r\n\r\nSubtract each term from the subsequent term to determine whether a common difference exists.\r\n<ol>\r\n \t<li>The sequence is not arithmetic because there is no common difference.\r\n<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>\r\n \t<li>The sequence is arithmetic because there is a common difference. The common difference is [latex]4[\/latex].\r\n<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>\r\n<\/ol>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nThe 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.\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\/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\" \/> Figure 1[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24930[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24931[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the regular interval between terms in a simple sequence and use it to write the sequence&#8217;s terms<\/li>\n<li>Use recursive and explicit formulas to represent and analyze arithmetic sequences<\/li>\n<\/ul>\n<\/section>\n<h2>Terms of an Arithmetic Sequence<\/h2>\n<p>Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">As an example, consider a woman who starts a small contracting business. She purchases a new truck for [latex]$25,000[\/latex]. After five years she estimates that she will be able to sell the truck for [latex]$8,000[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nThe loss in value of the truck will therefore be [latex]$17,000[\/latex], which is [latex]$3,400[\/latex] per year for five years.<br \/>\n[latex]\\\\[\/latex]<br \/>\nThe truck will be worth [latex]$21,600[\/latex] after the first year; [latex]$18,200[\/latex] after two years; [latex]$14,800[\/latex] after three years; [latex]$11,400[\/latex] after four years; and [latex]$8,000[\/latex] at the end of five years.<\/section>\n<p>The values of the truck in the example 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 [latex]\u20133,400[\/latex]. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and subtract [latex]3,400[\/latex] to find the subsequent term.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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.\" width=\"487\" height=\"68\" \/><figcaption class=\"wp-caption-text\">Arithmetic sequence<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>arithmetic sequence<\/h3>\n<div class=\"page\" title=\"Page 1074\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>An <strong>arithmetic sequence<\/strong> is a sequence where the difference between consecutive terms is always the same.<\/p>\n<\/div>\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\n<div class=\"column\">\n<p>The difference between consecutive terms, [latex]d[\/latex], and is called the <strong>common difference<\/strong>, for [latex]n[\/latex] greater than or equal to two.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The sequence below is another example of an arithmetic sequence. In this case the constant difference is [latex]3[\/latex]. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add [latex]3[\/latex] to find the subsequent term.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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.\" width=\"487\" height=\"68\" \/><figcaption class=\"wp-caption-text\">Arithmetic sequence<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q717238\">Show Solution<\/button><\/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 [latex]4[\/latex].\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<p><strong>Analysis of the Solution<\/strong><\/p>\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<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24930\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24930&theme=lumen&iframe_resize_id=ohm24930&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24931\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24931&theme=lumen&iframe_resize_id=ohm24931&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":363,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2695"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2695\/revisions"}],"predecessor-version":[{"id":7916,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2695\/revisions\/7916"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/363"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2695\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2695"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2695"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2695"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2695"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}