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Problems include proving divisibility and factor relationships, establishing statements involving exponents, and using an anchor step plus an inductive hypothesis and inductive step structure. Each item requests a complete induction argument showing the claim holds for all positive integers.","Kuta Software-Infinite Precalculus Name   \nMathematical Induction Date  Period   \nWrite the induction proof statements P􀀔 , Pk , and Pk 􀀎 􀀔 for each conjecture.  \n1) Pn: 􀀔􀀔 􀀎 􀀔􀀜 􀀎 􀀕􀀚 􀀎􀁽 􀀎 􀀛n 􀀎 􀀖 􀀠 n(􀀗n 􀀎 􀀚 )  \n2) Pn: 􀀚 n 􀀐 􀀗 n is divisible by 􀀖  \n3) Pn: 􀀗 n 􀁴 􀀗n  \nUse mathematical induction to prove that each statement is true for all positive integers  \n4) 􀀖􀀙 􀀎 􀀖􀀕􀀗 􀀎 􀀜􀀓􀀓 􀀎􀁽 􀀎 (􀀔􀀕n 􀀐 􀀙)􀀕 􀀠 􀀔􀀕n(􀀗n 􀀕 􀀐 􀀔)  \n5) 􀀖 is a factor of 􀀗 n 􀀎 􀀕  \n6) 􀀚 n 􀁴 􀀚n  \nKuta Software-Infinite Precalculus Name   \nMathematical Induction Date  Period   \nWrite the induction proof statements P􀀔 , Pk , and Pk 􀀎 􀀔 for each conjecture.  \n1) Pn: 􀀔􀀔 􀀎 􀀔􀀜 􀀎 􀀕􀀚 􀀎􀁽 􀀎 􀀛n 􀀎 􀀖 􀀠 n(􀀗n 􀀎 􀀚 )  \nP􀀔 : 􀀛 􀀎 􀀖 􀀠 􀀗 􀀎 􀀚  \nPk: 􀀔􀀔 􀀎 􀀔􀀜 􀀎 􀀕􀀚 􀀎􀁽 􀀎 􀀛k 􀀎 􀀖 􀀠 k(􀀗k 􀀎 􀀚)  \nPk 􀀎 􀀔 : 􀀔􀀔 􀀎 􀀔􀀜 􀀎 􀀕􀀚 􀀎􀁽 􀀎 􀀛k 􀀎 􀀖 􀀎 􀀛 (k 􀀎 􀀔) 􀀎 􀀖 􀀠 k(􀀗k 􀀎 􀀚) 􀀎 􀀛 (k 􀀎 􀀔) 􀀎 􀀖  \n2) Pn: 􀀚 n 􀀐 􀀗 n is divisible by 􀀖  \nP􀀔 : 􀀚 􀀔 􀀐 􀀗 􀀔 is divisible by 􀀖  \nPk: 􀀚 k 􀀐 􀀗 k is divisible by 􀀖 . Therefore, 􀀚 k 􀀐 􀀗 k 􀀠 􀀖 r for some integer r.  \nPk 􀀎 􀀔 : 􀀚 (k 􀀎 􀀔) 􀀐 􀀗 (k 􀀎 􀀔) is divisible by 􀀖  \n3) Pn: 􀀗 n 􀁴 􀀗n  \nP􀀔 : 􀀗 􀀔 􀁴 􀀗 􀂘 􀀔  \nPk: 􀀗 k 􀁴 􀀗k for some positive integer k 􀁴 􀀔 Pk 􀀎 􀀔 : 􀀗 k 􀀎 􀀔 􀁴 􀀗 (k 􀀎 􀀔)  \nUse mathematical induction to prove that each statement is true for all positive integers  \n4) 􀀖􀀙 􀀎 􀀖􀀕􀀗 􀀎 􀀜􀀓􀀓 􀀎􀁽 􀀎 (􀀔􀀕n 􀀐 􀀙)􀀕 􀀠 􀀔􀀕n(􀀗n 􀀕 􀀐 􀀔)  \nLet Pn be the statement 􀀖􀀙 􀀎 􀀖􀀕􀀗 􀀎 􀀜􀀓􀀓 􀀎􀁽 􀀎 (􀀔􀀕n 􀀐 􀀙)􀀕 􀀠 􀀔􀀕n(􀀗n 􀀕 􀀐 􀀔)  \nAP􀀔nicshtorrueStsepince (􀀔􀀕 􀀐 􀀙)􀀕 􀀠 􀀔􀀕 (􀀗 􀂘 􀀔 􀀕 􀀐 􀀔)  \nInductiAssumevtHypothesisat Pkis true: 􀀖􀀙 􀀎 􀀖􀀕􀀗 􀀎 􀀜􀀓􀀓 􀀎􀁽 􀀎 (􀀔􀀕k 􀀐 􀀙)􀀕 􀀠 􀀔􀀕k(􀀗k􀀕 􀀐 􀀔)  \nInductive Step  \nWe now show that Pk 􀀎 􀀔 is true:  \n􀀖􀀙 􀀎 􀀖􀀕􀀗 􀀎 􀀜􀀓􀀓 􀀎􀁽 􀀎 (􀀔􀀕k 􀀐 􀀙)􀀕 􀀎 (􀀔􀀕(k 􀀎 􀀔) 􀀐 􀀙)􀀕 = 􀀔􀀕k(􀀗k􀀕 􀀐 􀀔) 􀀎 (􀀔􀀕(k 􀀎 􀀔) 􀀐 􀀙)􀀕  \n􀀗􀀛 k􀀖 􀀐 􀀔􀀕k 􀀎 (􀀔􀀕k 􀀐 􀀙)􀀕  \n􀀗􀀛 k􀀖 􀀐 􀀔􀀕k 􀀎 􀀔􀀗􀀗k􀀕 􀀎 􀀔􀀗􀀗k 􀀎 􀀖􀀙  \n􀀗􀀛 k􀀖 􀀎 􀀔􀀗􀀗k􀀕 􀀎 􀀔􀀖􀀕k 􀀎 􀀖􀀙  \n􀀔􀀕 (􀀗k􀀖 􀀎 􀀔􀀕k􀀕 􀀎 􀀔􀀔 k 􀀎 􀀖)  \n􀀔􀀕 (􀀗k􀀖 􀀎 􀀔􀀕k􀀕 􀀎 􀀔􀀕k 􀀎 􀀗 􀀐 k 􀀐 􀀔)  \n􀀔􀀕 (􀀗(k􀀖 􀀎 􀀖 k􀀕 􀀎 􀀖 k 􀀎 􀀔) 􀀐 (k 􀀎 􀀔))  \n􀀔􀀕 (􀀗(k 􀀎 􀀔)􀀖 􀀐 (k 􀀎 􀀔))  \n􀀔􀀕 (k 􀀎 􀀔)(􀀗(k 􀀎 􀀔)􀀕 􀀐 􀀔)  \nConclusion  \nBy induction Pn is true for all n 􀁴 􀀔 .  \n5) 􀀖 is a factor of 􀀗 n 􀀎 􀀕  \nLet Pn be the statement 􀀖 is a factor of 􀀗 n 􀀎 􀀕  \nAnchor Step  \nP􀀔 is true: 􀀖 is a factor of 􀀗 􀀔 􀀎 􀀕  \nInductive Hypothesis  \nAssume that 􀀖 is a factor of 􀀗k 􀀎 􀀕 . Therefore, 􀀗 k 􀀎 􀀕 􀀠 􀀖 r for some integer r.  \nInductive Step  \nWe now show that Pk 􀀎 􀀔 is true: 􀀖 is a factor of 􀀗(k 􀀎 􀀔) 􀀎 􀀕  \n􀀗 􀂘 􀀗 k 􀀎 􀀕  \n(􀀖 􀀎 􀀔) 􀂘 􀀗 k 􀀎 􀀕  \n􀀖 􀂘 􀀗 k 􀀎 􀀗 k 􀀎 􀀕  \n􀀖 􀂘 􀀗 k 􀀎 􀀖 r 􀀖 (􀀗k 􀀎 r)  \nConclusion  \nBy induction Pn is true for all n 􀁴 􀀔 .  \n6) 􀀚 n 􀁴 􀀚n  \nLet Pn be the statement 􀀚 n 􀁴 􀀚n  \nAnchor Step  \nP􀀔 is true: 􀀚 􀀔 􀁴 􀀚 􀂘 􀀔  \nInductive Hypothesis  \nAssume that Pk is true: 􀀚 k 􀁴 􀀚k for some positive integer k 􀁴 􀀔  \nInductive Step  \nWe now show that Pk 􀀎 􀀔 is true:  \n􀀚 k 􀁴 􀀚k k 􀁴 􀀔  \n􀀚 􀂘 􀀚 k 􀁴 􀀚 􀂘 􀀚k 􀀗􀀕k 􀁴 􀀗􀀕 􀁴 􀀚  \n􀀚 k 􀀎 􀀔 􀁴 􀀗􀀜k  \n􀀚 k 􀀎 􀀔 􀁴 􀀚k 􀀎 􀀗􀀕k  \n􀀚 k 􀀎 􀀔 􀁴 􀀚k 􀀎 􀀚  \n􀀚 k 􀀎 􀀔 􀁴 􀀚 (k 􀀎 􀀔)  \nConclusion  \nBy induction Pn is true for all n 􀁴 􀀔 .  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaika4DuRfctGc","https://ap.wps.com/l/cbCaika4DuRfctGc","pdf",26880,"English","en",105,"# Mathematical Induction Worksheet\n## Conjecture 1 - Write Pn, Pk, and Pk + 1\n## Conjecture 2 - Divisibility Statement\n## Conjecture 3 - Exponent Inequality/Relation\n## Conjecture 4 - Induction Proof Setup and Inductive Step\n## Conjecture 5 - Factor Statement\n## Conjecture 6 - Exponent/Floor-Style Relation","[{\"question\":\"What induction proof statements are students asked to write for each conjecture?\",\"answer\":\"Students must write the induction proof statements Pn, Pk, and P(k + 1) (or an equivalent next-step statement) for each conjecture.\"},{\"question\":\"How does the worksheet structure a typical mathematical induction solution?\",\"answer\":\"Solutions use an anchor step (base case), an inductive hypothesis, and an inductive step to show the next case follows.\"},{\"question\":\"What kinds of conjectures are included in this worksheet?\",\"answer\":\"The worksheet includes conjectures about divisibility, factor relationships, and statements involving exponents, each to be proven for all positive integers.\"}]","Mathematical Induction - Infinite Precalculus Worksheets - Induction Proofs | PDF",1791427016,{"code":4,"msg":80,"data":81},"ok",{"site_id":74,"language":73,"slug":82,"title":65,"keywords":83,"description":66,"schema_data":84,"social_meta":138,"head_meta":140,"extra_data":142,"updated_unix":143},"mathematical-induction-infinite-precalculus-worksheets-induction-proofs","",{"@graph":85,"@context":137},[86,100,120],{"@type":87,"itemListElement":88},"BreadcrumbList",[89,93,95,98],{"item":90,"name":91,"@type":92,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":94,"name":9,"@type":92,"position":14},"https://docshare.wps.com/document/",{"item":96,"name":20,"@type":92,"position":97},"https://docshare.wps.com/document/exam/",3,{"item":99,"name":65,"@type":92,"position":19},"https://docshare.wps.com/document/mathematical-induction-infinite-precalculus-worksheets-induction-proofs/523561/",{"url":99,"name":65,"@type":101,"image":102,"author":107,"headline":65,"publisher":109,"fileFormat":112,"inLanguage":73,"description":66,"dateModified":113,"datePublished":114,"encodingFormat":112,"isAccessibleForFree":115,"interactionStatistic":116},"DigitalDocument",{"url":103,"@type":104,"width":105,"height":106},"https://docshare.wps.com/thumbnails/mathematical-induction-infinite-precalculus-worksheets-induction-proofs/523561.png","ImageObject",300,407,{"name":63,"@type":108},"Person",{"url":90,"name":110,"@type":111},"DocShare","Organization","application/pdf","2026-11-12","2026-10-08",true,{"@type":117,"interactionType":118,"userInteractionCount":24},"InteractionCounter",{"@type":119},"ViewAction",{"@type":121,"mainEntity":122},"FAQPage",[123,129,133],{"name":124,"@type":125,"acceptedAnswer":126},"What induction proof statements are students asked to write for each conjecture?","Question",{"text":127,"@type":128},"Students must write the induction proof statements Pn, Pk, and P(k + 1) (or an equivalent next-step statement) for each conjecture.","Answer",{"name":130,"@type":125,"acceptedAnswer":131},"How does the worksheet structure a typical mathematical induction solution?",{"text":132,"@type":128},"Solutions use an anchor step (base case), an inductive hypothesis, and an inductive step to show the next case follows.",{"name":134,"@type":125,"acceptedAnswer":135},"What kinds of conjectures are included in this worksheet?",{"text":136,"@type":128},"The worksheet includes conjectures about divisibility, factor relationships, and statements involving exponents, each to be proven for all positive integers.","https://schema.org",{"og:url":99,"og:type":139,"og:title":65,"og:site_name":110,"og:description":66},"article",{"robots":141,"canonical":99},"index,follow",{"doc_id":61,"site_id":74},1792536186]