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Unformatted text preview: 58 Student Solutions Hanual and Study Guide Formats for Proving Fonnulae by Mathematical Induction When usingmsthematical inductiontopcroveaibrmnla, students aresometimestemptedtopr=-==-? their proofs in a way that assumeswhat istobe proved. There are severalformsts you can b-idflfl the one shown most frequently in the textbook1 to avoid this fallacy. a crucial point is “it Ifyonarehopingtoprosethatsnequationiatruehntyouhaven’tyetdoneso,eithsrpretsoeI it with thewords “Wemustshowthat” ornutaqueetlonmarkahovethe equal sign. Fennel-t 1 {the fame: used most often in the textbook for the inductive stop): Start Fri theleft—haadsideflflfij oftheequationtobepmovedandauccessiwlytrsnsform it usingdefinitionsj known facts from basic algebra, and [for the inductive step} the inductive hypothesis until you oh =-'- the right-hand side {RES} of the equation. Formatflftheformotusedmostofieflmthe teatoookforthehosis step): Transform .._ LHSandthe RHSofthe equationtoheprmed independentimoneaflerthe other,until hothai-t: are shown to equal the same eaprassion. Because taro quantifies equal to the same quantity are” equaltoeaehother,youcsaeoneludethatthetwosidesoftheequationareequaltoeachother. Format .‘i': This format iajust like Format 2 accept that the computations are done in parallel. But; in order to avoid the failacy of assuming what is to be proved, do NOT put an equal sign between the twosides ofthe equation untiltheua-yiast step. Beparatethe twosidesofthe equation withs For-Inuit: ThisforrnatisjusterFbrmatE-exceptthstthetwosidesoftheeqnatiomaneseparated: byanequalsignwithaqueationmarkontop: ; Fonnnt 5: Start by wdtlng something tiles “We must shownr that"I and the equation you want to prove true. In aucoemive steps. indicate that this equation is true if, and only if, {or} various other equations are true. But he sure that both the directions of your "Hand only if” claims are correct. Inothes‘words, hesure that the s=dlrection telnetastrue oaths: direction. It'yonfinallyget downtoaneqnat-ionthat iaknown tohetnte. thenhecaneeeachauhsequenteqnation istrue if, and only if, the previous equation is true, you will have shown that the original equation is true. Example: Prose thatforsachintegsragl, . 1+3+5+,,,+{2n_1]=na 1—Thisisths equation. Prodthat the equation istrnafor n. = 1: Solution {annotfifi Whennzl,theLI-l3oftheequationequalsl,andtheRHEiequalsizsrhichaisoequalsl.Sothe equationistruefiorfl=L Solution (mt 5): thnn=2.wemustshowthat1=1”.Bwausethisistrue,theeqnationiatmeforn=1. ...
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