[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573077-105":53,"doc-detail-573077-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-cheat-sheet","Sequences and Series - Cheat Sheet","","Sequences and series cheat sheet defining core concepts and terminology used in arithmetic and geometric progressions. Covers what makes a sequence increasing, decreasing, or periodic, and explains how arithmetic terms use a common difference. Derives key formulas for the nth term and for the sum of the first n terms, including a proof method. Then introduces geometric sequences, their nth term and partial sums, explains convergence versus divergence using the ratio condition, and presents the sum to infinity. Includes solved examples and a brief introduction to recurrence relations.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/sequences-and-series-cheat-sheet/573077/",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-cheat-sheet/573077.png","ImageObject",442,249,{"name":88,"@type":89},"Valentina","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"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. A series is the sum of the terms of a sequence.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How do you identify an arithmetic sequence and find its nth term?",{"text":113,"@type":109},"An arithmetic sequence has a common difference between consecutive terms. Its nth term uses the first term and the common difference to express the term index n.",{"name":115,"@type":106,"acceptedAnswer":116},"When does the sum to infinity of a geometric sequence exist?",{"text":117,"@type":109},"The sum to infinity exists only for convergent geometric sequences, which happen exactly when the absolute value of the common ratio satisfies |r| \u003C 1.","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},573077,1792033555,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":76,"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},13056703020460,"https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923","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:  ","cbCaiimZxQvCAaxA","https://ap.wps.com/l/cbCaiimZxQvCAaxA","pdf",1666147,"English","# Definitions\n## Sequence vs. series\n## Increasing, decreasing, periodic\n# Arithmetic sequences and series\n## Common difference and nth term\n## Sum of the first n terms (with proof method)\n## Example applications\n# Geometric sequences and series\n## Common ratio and nth term\n## Partial sum formulas (with proof method)\n## Convergence vs. divergence\n## Sum to infinity\n# Recurrence relations\n## Definition and how to generate a sequence\n## Example setup","[{\"question\":\"What is the difference between a sequence and a series?\",\"answer\":\"A sequence is a list of terms. A series is the sum of the terms of a sequence.\"},{\"question\":\"How do you identify an arithmetic sequence and find its nth term?\",\"answer\":\"An arithmetic sequence has a common difference between consecutive terms. Its nth term uses the first term and the common difference to express the term index n.\"},{\"question\":\"When does the sum to infinity of a geometric sequence exist?\",\"answer\":\"The sum to infinity exists only for convergent geometric sequences, which happen exactly when the absolute value of the common ratio satisfies |r| \\u003c 1.\"}]","Sequences and Series - Cheat Sheet | PDF",1791931864]