{"id":2677,"date":"2025-08-13T18:26:00","date_gmt":"2025-08-13T18:26:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2677"},"modified":"2025-09-24T19:52:09","modified_gmt":"2025-09-24T19:52:09","slug":"sequences-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sequences-background-youll-need-1\/","title":{"raw":"Sequences: Background You'll Need 1","rendered":"Sequences: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Extend numerical patterns<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>numerical pattern<\/h3>\r\nA numerical pattern is a sequence of numbers that follows a specific rule or relationship.\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">To extend a pattern, we identify the rule and use it to find the next terms. Common patterns include adding or subtracting the same number (arithmetic), multiplying or dividing by the same number (geometric), or following more complex rules.<\/p>\r\n\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Finding the Pattern<\/h2>\r\n<p class=\"whitespace-normal break-words\">The first step in extending any numerical pattern is identifying what rule connects the numbers.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Identify the pattern and find the next three terms: 5, 8, 11, 14, ...[reveal-answer q=\"413486\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"413486\"][latex] \\begin{aligned} 8 - 5 &amp;= 3 \\ 11 - 8 &amp;= 3 \\ 14 - 11 &amp;= 3 \\end{aligned} [\/latex] The pattern adds 3 to each term to get the next term. Next three terms: 17, 20, 23[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312668[\/ohm_question]<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Common Types of Patterns<\/h2>\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>Question Help: Steps to Find Numerical Patterns<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Find the difference between each term using subtraction. Does the pattern add the same thing every time? Or add a consistently changing amount?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the ratio between each term using division. (Divide the second term by the first, the third time by the second, etc). Does the ratio stay the same every time?<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Extend numerical patterns<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>numerical pattern<\/h3>\n<p>A numerical pattern is a sequence of numbers that follows a specific rule or relationship.<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">To extend a pattern, we identify the rule and use it to find the next terms. Common patterns include adding or subtracting the same number (arithmetic), multiplying or dividing by the same number (geometric), or following more complex rules.<\/p>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Finding the Pattern<\/h2>\n<p class=\"whitespace-normal break-words\">The first step in extending any numerical pattern is identifying what rule connects the numbers.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Identify the pattern and find the next three terms: 5, 8, 11, 14, &#8230;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q413486\">Show Answer<\/button><\/p>\n<div id=\"q413486\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} 8 - 5 &= 3 \\ 11 - 8 &= 3 \\ 14 - 11 &= 3 \\end{aligned}[\/latex] The pattern adds 3 to each term to get the next term. Next three terms: 17, 20, 23<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312668\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312668&theme=lumen&iframe_resize_id=ohm312668&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Common Types of Patterns<\/h2>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>Question Help: Steps to Find Numerical Patterns<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Find the difference between each term using subtraction. Does the pattern add the same thing every time? Or add a consistently changing amount?<\/li>\n<li class=\"whitespace-normal break-words\">Find the ratio between each term using division. (Divide the second term by the first, the third time by the second, etc). Does the ratio stay the same every time?<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2677"}],"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\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2677\/revisions"}],"predecessor-version":[{"id":4341,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2677\/revisions\/4341"}],"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\/2677\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2677"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2677"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2677"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2677"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}