{"id":158,"date":"2025-02-13T22:44:26","date_gmt":"2025-02-13T22:44:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-2\/"},"modified":"2026-03-25T21:40:42","modified_gmt":"2026-03-25T21:40:42","slug":"arithmetic-sequences-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-2\/","title":{"raw":"Arithmetic Sequences: Learn It 1","rendered":"Arithmetic Sequences: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the common difference for an arithmetic sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Write the formula for an arithmetic sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use arithmetic sequences to solve realistic scenarios<\/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<img class=\"aligncenter wp-image-4977\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy-300x54.png\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" width=\"555\" height=\"100\" \/>\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.<img class=\"aligncenter wp-image-4978 \" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram-300x52.png\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" width=\"461\" height=\"80\" \/>\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<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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321958[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321959[\/ohm_question]<\/section><\/div>\r\n<dl id=\"fs-id1165137415248\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135174993\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find the common difference for an arithmetic sequence.<\/li>\n<li style=\"font-weight: 400;\">Write the formula for an arithmetic sequence.<\/li>\n<li style=\"font-weight: 400;\">Use arithmetic sequences to solve realistic scenarios<\/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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4977\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy-300x54.png\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" width=\"555\" height=\"100\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy-300x54.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy-65x12.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy-225x40.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy-350x63.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235748\/11.2.L1.Diagram-copy.png 582w\" sizes=\"(max-width: 555px) 100vw, 555px\" \/><\/p>\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.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4978\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram-300x52.png\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" width=\"461\" height=\"80\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram-300x52.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram-65x11.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram-225x39.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram-350x61.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02235825\/11.2.L1.Diagram.png 472w\" sizes=\"(max-width: 461px) 100vw, 461px\" \/><\/p>\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<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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321958\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321958&theme=lumen&iframe_resize_id=ohm321958&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321959\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321959&theme=lumen&iframe_resize_id=ohm321959&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<dl id=\"fs-id1165137415248\" class=\"definition\">\n<dd id=\"fs-id1165135174993\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":8,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"module-header":"learn_it","content_attributions":[{"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":""}],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/158"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/158\/revisions"}],"predecessor-version":[{"id":6024,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/158\/revisions\/6024"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/156"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/158\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=158"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=158"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=158"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}