[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-393805-105":59,"doc-detail-393805-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","summations-by-formulas-79-exam-questions-basic-questions-17","Summations by Formulas - 79 Exam Questions - Basic Questions - 17","","A set of 79 exam questions focused on summation techniques using standard formulas. The problems range from evaluating specific sums to proving identities involving polynomial terms multiplied by powers of r or combinations of r with shifted factors. Several tasks ask for fully factorized expressions of summation results, including expressions for sums of the first n terms of arithmetic-like series and sums defined by congruent quadratic forms, with parameters to be determined as positive integers.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/summations-by-formulas-79-exam-questions-basic-questions-17/393805/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/summations-by-formulas-79-exam-questions-basic-questions-17/393805.png","ImageObject",300,407,{"name":92,"@type":93},"moss~","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What skills do these questions test?","Question",{"text":113,"@type":114},"They test the ability to use standard summation results to evaluate sums, prove polynomial summation identities, and convert results into fully factorized forms.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Do the problems include proof-based tasks?",{"text":118,"@type":114},"Yes. Several questions explicitly ask to “show” using standard summation results, including determining integer parameters in the resulting closed forms.",{"name":120,"@type":111,"acceptedAnswer":121},"What type of final answers are required?",{"text":122,"@type":114},"Many questions require expressions in fully simplified factorized form, especially for sums of the first n terms and for general summation expressions depending on n (or other indices).","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},393805,1790528213,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":44,"preview_urls":147},962090895452,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Created by T. Madas  \nSERIES  \n79 EXAM QUESTIONS  \nCreated by T. Madas  \nCreated by T. Madas  \nSUMMATIONS BY FORMULAS  \n17 BASIC QUESTIONS  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 1 (**)  \nUse standard results on summations to find the value of  \n (r −1)(3r − 2 ) .  \nr=36  \nFP1-B , 66638  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 2 (**)  \nUse standard results on summations to show that  \n􀀅n r (r +1)(r + 5 ) = ~~ ~~14 n (n + a )(n + b )(n + c ) ,  \nr=1  \nwhere a , b , and c are positive integers to be found.  \n\n| FP1-H | , | a = 1, b = 2, c = 7 |\n| --- | --- | --- |\n\nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 3 (**)  \nUse standard results on summations to show that  \n􀀅n r 2 (r −1) = 112 n (n −1)(n +1)(3n + 2 ) + m ,  \nr=1  \nwhere m is an integer to be found.  \nSYNF-X , m = −22  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 4 (**)  \nUse standard results on summations to show that  \n􀀅n r 3 (r +1)(r −1) = ~~1~~6 n 2 (n +1)2 (n −1)(n + 2 ) .  \nr=1  \nYou may assume without proof that 􀀅n r 5 = 1~~1~~2 n 2 (n +1)2 ( 2n2 + 2n −1)  \nr=1  \nSYNF-Z , proof  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 5 (**)  \nF (r ) ≡ 􀀅r n (n −1)(n + 2 ) .  \nn=1  \nUse standard results on summations express F (n ) in fully factorized from.  \n\n| SYNF-M  , | F (r ) = ~~ ~~1~~1~~2 r (r +1)(r −1)(3r +10 ) |\n| --- | --- |\n\nQuestion 6 (**+)  \nFind, in fully simplified factorized form, an expression for the sum of the first n terms of the following series.  \n(5× 3) + (11× 7 ) + (17 ×11) + (23×15 )+ ...  \nFP1-C , n2 (8n + 7 )  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 7 (**+)  \nShow by using standard summation results that …  \na) … 􀀅n (r +1)(r + 5 ) = ~~ ~~16 n (n + 7 )(2n + 7 ) .  \nr=1  \nb)… (r +1)(r + 5 ) = 26495 .  \nr=11  \nSYNF-C , proof  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 8 (**+)  \nShow by using standard summation results that …  \na) … 􀀅n ( k 2 − k −1) = ~~ ~~13n (n + 2 )(n − 2 ) .  \nk =1  \nb)… ( k 2 − k −1) = 21049 .  \nk =10  \nFP1-G , proof  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 9 (**+)  \nFind, in fully factorized form, an expression for the sum  \n( p 3 + p2 ) . p=1  \n\n| \u003Cbr>, |  ~~ ~~1~~1~~2 k (k +1)(k + 2 )(3k +1)  |\n| --- | --- |\n\nQuestion 10 (**+)  \nFind, in fully factorized form, an expression for the sum  \n( 3r2 − ~~ 1~~2) . r=1  \nCreated by T. Madas","cbCaibWa6TUYUM6T","https://ap.wps.com/l/cbCaibWa6TUYUM6T","pdf",13318624,84,"English","# Series\n## Summations by formulas\n### Question 1\n### Question 2\n### Question 3\n### Question 4\n### Question 5\n### Question 6\n### Question 7\n### Question 8\n### Question 9\n### Question 10","[{\"question\":\"What skills do these questions test?\",\"answer\":\"They test the ability to use standard summation results to evaluate sums, prove polynomial summation identities, and convert results into fully factorized forms.\"},{\"question\":\"Do the problems include proof-based tasks?\",\"answer\":\"Yes. Several questions explicitly ask to “show” using standard summation results, including determining integer parameters in the resulting closed forms.\"},{\"question\":\"What type of final answers are required?\",\"answer\":\"Many questions require expressions in fully simplified factorized form, especially for sums of the first n terms and for general summation expressions depending on n (or other indices).\"}]","Summations by Formulas - 79 Exam Questions - Basic Questions - 17 | PDF",1790393519,212,[148,152,156,160,164,168,172,176,180],{"page_no":8,"url":149,"width":150,"height":151},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/c89657f88fd7a66eddb3efc1f409f088.webp",983,1391,{"page_no":14,"url":153,"width":154,"height":155},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/2548b11f7123ee0dbcd3dc96a0221c62.webp",982,1389,{"page_no":81,"url":157,"width":158,"height":159},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/952a4243614311a28d93fc827fece83b.webp",981,1388,{"page_no":19,"url":161,"width":162,"height":163},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/243b35ad149bcf21e41bc3435abe6e9e.webp",980,1386,{"page_no":24,"url":165,"width":166,"height":167},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/9545a7029f8aa190a1db8282479b8065.webp",979,1385,{"page_no":29,"url":169,"width":170,"height":171},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/e4a7195675507e609b22af275c67e128.webp",978,1383,{"page_no":34,"url":173,"width":174,"height":175},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/24a0c96f271ab6c0f91ba91f2ca0a769.webp",977,1382,{"page_no":39,"url":177,"width":178,"height":179},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/7f8cddd502ec85121ef575e90cff342a.webp",976,1381,{"page_no":44,"url":181,"width":182,"height":183},"https://res-academy.cache.wpscdn.com/upload/kdocshare/2026-11-08/8ac1f2600b58997cc4755679778ddb11.webp",975,1379]