[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572266-105":53,"doc-detail-572266-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","sequences-and-series-chapter-3-cheat-sheet","Sequences and Series - Chapter 3 - Cheat Sheet","","Cheat sheet summarizing key ideas in sequences and series, focusing on definitions and core formulae. Covers increasing, decreasing, and periodic sequences; arithmetic sequences and series via common difference, nth-term, and sum formulas with a proof approach; and geometric sequences and series via common ratio, nth-term and sum to n terms, including proof steps and conditions for convergence. Includes guidance on divergent vs convergent infinite sums and introduces recurrence relations with an example.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/sequences-and-series-chapter-3-cheat-sheet/572266/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/sequences-and-series-chapter-3-cheat-sheet/572266.png","ImageObject",442,249,{"name":88,"@type":89},"lilac","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is the difference between a sequence and a series?","Question",{"text":108,"@type":109},"A sequence is a list of terms, while a series is the sum of those terms. Sequence terms are presented separately, whereas series terms are added together.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How do you identify an arithmetic sequence and compute its sum?",{"text":113,"@type":109},"An arithmetic sequence has a common difference between consecutive terms. The nth-term and the sum of the first n terms are given by standard arithmetic formulas, which the sheet also shows how to prove.",{"name":115,"@type":106,"acceptedAnswer":116},"When does the infinite sum of a geometric series exist?",{"text":117,"@type":109},"The infinite sum exists only for convergent geometric sequences, which occur exactly when the absolute value of the common ratio is less than 1. If the common ratio equals 1, the finite-n formula becomes invalid due to division by zero.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572266,1792053429,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},3985747861865,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Sequences and Series Cheat Sheet  \nA sequence is a list of terms. For example, 3, 6, 9, 12, 15,…  \nA series is the sum of a list of terms. For example, 3 + 6 + 9 + 12 + 15 + …  \nThe terms of a sequence are separated by a comma, while with a series they are all added together.  \nHere are some important definitions prefacing the content in this chapter:  \n▪ A sequence is increasing if each term is greater than the previous.  \ne.g. 4, 9, 14, 19,…  \n▪ A sequence is decreasing if each term is less than the previous.  \ne.g. 5, 4, 3, 2, 1,…  \n▪ A sequence is periodic if the terms repeat in a cycle; 􀝑􀯡+􀯞 = 􀝑􀯡 for some k, which is known as the order of the sequence. e.g. -3, 1, -3, 1, -3,… is periodic with order 2.  \nAn arithmetic sequence is one where there is a common difference between each term. Arithmetic sequences are of the form  \n􀜽, 􀜽 + 􀝀, 􀜽 + 2􀝀, 􀜽 + 3􀝀, …  \nwhere 􀜽 is the first term and 􀝀 is the common difference.  \n• The nth term of an arithmetic series is given by: 􀢛􀢔 = 􀢇 + (􀢔 − 􀫚)􀢊  \nFactorising out 􀜵􀯡 from the LHS  \n and 􀜽 from the RHS  \nAn arithmetic series is the sum of the terms of an arithmetic sequence.  \n• The sum of the first 􀝊 terms of an arithmetic series is given by 􀡿􀢔 = 􀢔􀫛 [􀫛􀢇 + (􀢔 − 􀫚)􀢊] or  \n􀡿􀢔 = 􀢔􀫛 (􀢇 + 􀢒)  \nwhere 􀜽 is the first term, 􀝀 is the common difference and 􀝈 is the last term.  \nYou need to be able to prove this result. Here is the proof:  \nExample 1: Prove that the sum of the first n terms of an arithmetic series is 􀜵􀯡 = 􀯡2 [2􀜽 + ( 􀝊 − 1)􀝀] .  \nWe start by writing the sum out normally [1], and then in reverse [2]:  \n[1] 􀜵􀯡 = 􀜽 + ( 􀜽 + 􀝀) + ( 􀜽 + 2􀝀) + ⋯ + ( 􀜽 + ( 􀝊 − 2)􀝀) + ( 􀜽 + ( 􀝊 − 1)􀝀)  \n[2] 􀜵􀯡 = ( 􀜽 + ( 􀝊 − 1)􀝀) + ( 􀜽 + ( 􀝊 − 2)􀝀) + ⋯ + ( 􀜽 + 2􀝀) + ( 􀜽 + 􀝀) + 􀜽 Adding [1] and [2] gives us:  \n[1] + [2]: 2􀜵􀯡 = 􀝊( 2􀜽 + ( 􀝊 − 1)􀝀)  \n􀝊  \n∴ 􀜵􀯡 = [2􀜽 + ( 􀝊 − 1)􀝀]  \n2  \nExample 2: The fifth term of an arithmetic series is 33. The tenth term is 68. The sum of the first n  \nterms is 2225. Show that 7􀝊2 + 3􀝊 − 4450 = 0, and hence find the value of n.  \n5􀝐ℎ 􀝐􀝁􀝎􀝉 􀝅􀝏 33 ∴ 􀜽 + 4􀝀 = 33 [1] (nth term formula)  \n10􀝐ℎ 􀝐􀝁􀝎􀝉 􀝅􀝏 68 ∴ 68 = 􀜽 + 9􀝀 [2] (nth term formula)  \nSolving [1] and [2] simultaneously, we find that 􀝀 = 7 and 􀜽 = 5. Sum of first n terms is 2225 ∴ 􀯡2 [2􀜽 + ( 􀝊 − 1)􀝀] = 2225  \n􀝊2 [10 + ( 􀝊 − 1)(7)] = 2225  \n􀝊 ( 3 + 7􀝊) = 2225( 2)  \n7􀝊2 + 3􀝊 − 4450 = 0  \nTo find the value of n, we just need to solve the quadratic. Using the quadratic formula, we find that 􀝊 = 25 􀝋􀝎 􀝊 = −25 .4. Since the term number must be a positive integer, we can conclude that 􀝊 = 25.  \nThe defining feature of a geometric sequence is that you must multiply by a common ratio, r, to get from one term to the next. Geometric sequences are of the form  \n􀜽, 􀜽􀝎, 􀜽􀝎 2, 􀜽􀝎 3, 􀜽􀝎4, …  \nwhere 􀜽 is the first term in the sequence and 􀝎 is the common ratio.  \n• The nth term of a geometric sequence is given by: 􀢛􀢔 = 􀢇􀢘􀢔−􀫚  \nIt can help in many questions to use the fact that 􀯨􀳖􀯨1 = 􀯨􀯨􀳖􀳖21 = 􀝎 . This is especially helpful when the terms of the sequence are given in terms of an unknown constant. Part a of example 4 highlights this.  \nA geometric series is the sum of the terms of a geometric sequence.  \n• The sum of the first n terms of a geometric series is given by:  \n􀡿􀢔 = 􀢇~~ ~~(􀫚􀫚~~ ~~−~~ ~~􀢘􀢘􀢔~~ ~~)  \nby multiplying the top and bottom of the fraction by-1, we can also use  \n􀢇 (􀢘􀢔 − 􀫚)  \n􀡿􀢔 = 􀢘 − 􀫚  \nYou need to be able to prove this result. Here is the proof:  \nExample 3: Prove that the sum of the first n terms of a geometric series is 􀜵􀯡 = 􀯔(11−􀯥􀯥􀳙~~ ~~)  \n􀜵􀯡 = 􀜽 + 􀜽􀝎 + 􀜽􀝎2 + ⋯ + 􀜽􀝎􀯡−1 [1]  \nmultiplying the sum by 􀝎 􀝎􀜵􀯡 = 􀜽􀝎 + 􀜽􀝎2 + 􀜽􀝎3 + ⋯ + 􀜽􀝎􀯡 [2]  \nSubtracting [2] from [1] 􀜵􀯡 − 􀝎􀜵􀯡 = 􀜽 − 􀜽􀝎􀯡  \n⇒ 􀜵􀯡 ( 1 − 􀝎) = 􀜽 ( 1 − 􀝎􀯡 ) Factoring out 􀜵􀯡 and 􀜽  \n∴ 􀜵􀯡 = 􀯔(11−􀯥􀯥􀳙~~ ~~) Dividing by 1 − 􀝎  \nSince division by zero is undefined, this formula is invalid when 􀝎 = 1.  \nThe sum to infinity of a geometric sequence is the sum of the first n terms as n approaches infinity. This does not exist for all geometric sequences. Let’s look at two examples:  ","cbCaiqBaMaMyp5z8","https://ap.wps.com/l/cbCaiqBaMaMyp5z8","pdf",2329950,"English","# Arithmetic sequences and series\n## Definitions and order\n## nth term and sum of first n terms\n# Geometric sequences and series\n## nth term and sum of first n terms\n## Infinite sums and convergence\n# Recurrence relations\n## Definition and example","[{\"question\":\"What is the difference between a sequence and a series?\",\"answer\":\"A sequence is a list of terms, while a series is the sum of those terms. Sequence terms are presented separately, whereas series terms are added together.\"},{\"question\":\"How do you identify an arithmetic sequence and compute its sum?\",\"answer\":\"An arithmetic sequence has a common difference between consecutive terms. The nth-term and the sum of the first n terms are given by standard arithmetic formulas, which the sheet also shows how to prove.\"},{\"question\":\"When does the infinite sum of a geometric series exist?\",\"answer\":\"The infinite sum exists only for convergent geometric sequences, which occur exactly when the absolute value of the common ratio is less than 1. If the common ratio equals 1, the finite-n formula becomes invalid due to division by zero.\"}]","Sequences and Series - Chapter 3 - Cheat Sheet | PDF",1791926492]