[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522665-105":59,"doc-detail-522665-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","geometric-sequences-common-ratio-and-explicitrecursive-formulas-worksheet","Geometric Sequences - Common Ratio and Explicit/Recursive Formulas Worksheet","","Kuta Software-Infinite Algebra 2 worksheet on geometric sequences covering both conceptual identification and formula-based computations. Students determine whether given sequences are geometric and find the common ratio, then use explicit formulas to compute the first five terms and specific terms such as the 8th. Additional problems require working from recursive rules to obtain the common ratio, first terms, and the explicit form, and solving using given terms to derive later terms and recursion.",{"@graph":69,"@context":122},[70,84,105],{"@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/geometric-sequences-common-ratio-and-explicitrecursive-formulas-worksheet/522665/",{"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/geometric-sequences-common-ratio-and-explicitrecursive-formulas-worksheet/522665.png","ImageObject",300,407,{"name":92,"@type":93},"Margaret","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you determine whether a sequence is geometric?","Question",{"text":112,"@type":113},"Check whether the ratio between consecutive terms is constant. If the common ratio stays the same, the sequence is geometric; otherwise it is not.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you find the common ratio for a geometric sequence?",{"text":117,"@type":113},"Compute the ratio of any term to the previous term, then verify the same ratio works across consecutive terms.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the difference between explicit and recursive formulas in these problems?",{"text":121,"@type":113},"The explicit formula gives a direct expression for a_n in terms of n, while the recursive formula defines a_n using a_{n-1}. The worksheet asks students to find both forms depending on the given information.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},522665,1791732451,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52,"preview_page_count":4},137451207643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Kuta Software-Infinite Algebra 2 Name   \nGeometric Sequences Date  Period   \nDetermine if the sequence is geometric. If it is, find the common ratio.  \n1) −1, 6, −36, 216, ... 2) −1, 1, 4, 8, ...  \n3) 4, 16, 36, 64, ... 4) −3, −15, −75, −375, ...  \n5) −2, −4, −8, −16, ... 6) 1, −5, 25, −125, ...  \nGiven the explicit formula for a geometric sequence find the first five terms and the 8th term.  \n7) an = 3 n − 1 8) an = 2 ⋅ (14) n − 1  \n9) a = −2.5 ⋅ 4 n − 1 10) a = −4 ⋅ 3 n − 1 n n  \nGiven the recursive formula for a geometric sequence find the common ratio, the first five terms, and the explicit formula.  \n11) an = an − 1 ⋅ 2 12) an = an − 1 ⋅ −3 a 1 = 2 a 1 = −3  \n13) an = an − 1 ⋅ 5 14) an = an − 1 ⋅ 3 a 1 = 2 a 1 = −3  \nGiven the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula.  \n15) a 1 = 0.8, r = −5 16) a 1 = 1, r = 2  \nGiven the first term and the common ratio of a geometric sequence find the recursive formula and the three terms in the sequence after the last one given.  \n17) a 1 = −4, r = 6 18) a 1 = 4, r = 6  \n19) a 1 = 2, r = 6 20) a 1 = −4, r = 4  \nGiven a term in a geometric sequence and the common ratio find the first five terms, the explicit formula, and the recursive formula.  \n21) a4 = 25, r = −5 22) a 1 = 4, r = 5  \nGiven two terms in a geometric sequence find the 8th term and the recursive formula.  \n23) a4 = −12 and a5 = −6 24) a5 = 768 and a2 = 12  \n25) a 1 = −2 and a5 = −512 26) a5 = 3888 and a3 = 108  \nKuta Software-Infinite Algebra 2 Name   \nGeometric Sequences Date  Period   \nDetermine if the sequence is geometric. If it is, find the common ratio.  \n1) −1, 6, −36, 216, ... 2) −1, 1, 4, 8, ...  \nr = −6 Not geometric  \n3) 4, 16, 36, 64, ... 4) −3, −15, −75, −375, ...  \nNot geometric r = 5  \n5) −2, −4, −8, −16, ... 6) 1, −5, 25, −125, ...  \nr = 2 r = −5  \nGiven the explicit formula for a geometric sequence find the first five terms and the 8th term.  \n7)  \n9)  \na = 3 n − 1  \nn  \nFirst Five Terms: 1, 3, 9, 27, 81  \na8 = 2187  \na = −2.5 ⋅ 4 n − 1  \nn  \nFirst Five Terms: −2.5, −10, −40, −160, −640 a8 = −40960  \n8) an = 2 ⋅ (14) n − 1    \n1 1 1 1 First Five Terms: 2, , , ,  \n2 8 32 128  \na8 = ~~ ~~81192  \n10) a = −4 ⋅ 3 n − 1  \nn  \nFirst Five Terms: −4, −12, −36, −108, −324 a8 = −8748  \nGiven the recursive formula for a geometric sequence find the common ratio, the first five terms, and the explicit formula.  \n11) an = an − 1 ⋅ 2  \na 1 = 2  \nCommon Ratio: r = 2  \nFirst Five Terms: 2, 4, 8, 16, 32  \nExplicit: a = 2 ⋅ 2 n − 1  \nn  \n13) an = an − 1 ⋅ 5  \na 1 = 2  \nCommon Ratio: r = 5  \nFirst Five Terms: 2, 10, 50, 250, 1250  \nn − 1  \nExplicit: a = 2 ⋅ 5  \nn  \n12) an = an − 1 ⋅ −3 a 1 = −3  \nCommon Ratio: r = −3  \nFirst Five Terms: −3, 9, −27, 81, −243  \nExplicit: a = −3 ⋅ (−3)n − 1  \nn  \n14) an = an − 1 ⋅ 3  \na 1 = −3  \nCommon Ratio: r = 3  \nFirst Five Terms: −3, −9, −27, −81, −243  \nn − 1  \nExplicit: a = −3 ⋅ 3  \nn  \nGiven the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula.  \n15) a 1 = 0.8, r = −5 16) a 1 = 1, r = 2  \nFirst Five Terms: 0.8, −4, 20, −100, 500 First Five Terms: 1, 2, 4, 8, 16  \nExplicit: a = 0.8 ⋅ (−5)n − 1 Explicit: a = 2n − 1  \nn n  \nGiven the first term and the common ratio of a geometric sequence find the recursive formula and the three terms in the sequence after the last one given.  \n17) a 1 = −4, r = 6  \nNext 3 terms: Recursive: a  \nna 1  \n19) a 1 = 2, r = 6  \nNext 3 terms: Recursive: a  \nna 1  \n18) a 1 = 4, r = 6  \n−24, −144, −864 Next 3 terms: 24, 144, 864  \n= an − 1 ⋅ 6 Recursive: an = an − 1 ⋅ 6  \n= −4 a 1 = 4  \n20) a 1 = −4, r = 4  \n12, 72, 432 Next 3 terms: −16, −64, −256  \n= an − 1 ⋅ 6 Recursive: an = an − 1 ⋅ 4  \n= 2 a 1 = −4  \nGiven a term in a geometric sequence and the common ratio find the first five terms, the explicit formula, and the recursive formula.  \n21) a4 = 25, r = −5  \nFirst Five Terms: −0.2, 1, −5, 25, −125  \nExplicit: a = −0.2 ⋅ (−5)n − 1  \nn  \nRecurs","cbCaijgBsgcdaRVg","https://ap.wps.com/l/cbCaijgBsgcdaRVg","pdf",38063,"English","# Determine Geometric Sequences\n## Common Ratio and Geometric Check\n# Explicit Formula Problems\n## First Five Terms and the 8th Term\n# Recursive Formula Problems\n## Common Ratio, First Terms, and Explicit Form\n# Problems Given Terms\n## Find Later Terms and Recursive Formula","[{\"question\":\"How do you determine whether a sequence is geometric?\",\"answer\":\"Check whether the ratio between consecutive terms is constant. If the common ratio stays the same, the sequence is geometric; otherwise it is not.\"},{\"question\":\"How do you find the common ratio for a geometric sequence?\",\"answer\":\"Compute the ratio of any term to the previous term, then verify the same ratio works across consecutive terms.\"},{\"question\":\"What is the difference between explicit and recursive formulas in these problems?\",\"answer\":\"The explicit formula gives a direct expression for a_n in terms of n, while the recursive formula defines a_n using a_{n-1}. The worksheet asks students to find both forms depending on the given information.\"}]","Geometric Sequences - Common Ratio and Explicit/Recursive Formulas Worksheet | PDF",1791426035]