Principle of Mathematical Induction | Introduction, Steps and ...

Mathematical Induction is a technique of proving a statement, theorem or formula which is thought to be true, for each and every natural number n. MathsMathArticlePrincipleOfMathematicalInductionLearnExamples PrincipleofMathematicalInduction MathematicalInductionisatechniqueofproving astatement,theoremorformulawhichisthoughttobetrue,foreachandeverynaturalnumbern.Bygeneralizingthisinformofaprinciplewhichwewouldusetoproveanymathematicalstatementis‘PrincipleofMathematicalInduction‘. Forexample:13+23 +33+…..+n3=(n(n+1)/2)2,thestatementisconsideredhereastrueforallthevaluesofnaturalnumbers. UnderstandingMathematicalInductionWithExamples ImportantQuestionsClass11MathsChapter4PrinciplesMathematicalInduction PrincipleofMathematicalInductionSolutionandProof ConsiderastatementP(n),wherenisanaturalnumber.ThentodeterminethevalidityofP(n)foreveryn,usethefollowingprinciple: Step1: Checkwhetherthegivenstatementistrueforn=1. Step2:AssumethatgivenstatementP(n)isalsotrueforn=k,wherekisanypositiveinteger. Step3: ProvethattheresultistrueforP(k+1)foranypositiveintegerk. Iftheabove-mentionedconditionsaresatisfied,thenitcanbeconcludedthatP(n)istrueforallnnaturalnumbers. Proof: Thefirststepoftheprincipleisafactualstatementandthesecondstepisaconditionalone.AccordingtothisifthegivenstatementistrueforsomepositiveintegerkonlythenitcanbeconcludedthatthestatementP(n)isvalidforn=k+1. ThisisalsoknownastheinductivestepandtheassumptionthatP(n)istrueforn=kisknownastheinductivehypothesis. Solvedproblems Example1: Provethatthesumofcubesofnnaturalnumbersisequalto(n(n+1)2)2 forallnnaturalnumbers. Solution: Inthegivenstatementweareaskedtoprove: 13+23+33+⋯+n3 =(n(n+1)2)2 Step1:NowwiththehelpoftheprincipleofinductioninmathletuscheckthevalidityofthegivenstatementP(n)forn=1. P(1)=(1(1+1)2)2 =1Thisistrue. Step2:Nowasthegivenstatementistrueforn=1weshallmoveforwardandtryprovingthisforn=k,i.e., 13+23+33+⋯+k3=(k(k+1)2)2 . Step3:LetusnowtrytoestablishthatP(k+1)isalsotrue.                   13+23+33+⋯+k3+(k+1)3 =(k(k+1)2)2+(k+1)3                  ⇒13+23+33+⋯+k3+(k+1)3=k2(k+1)4+(k+1)3                  =k2(k+1)2+4((k+1)3)4                  =(k+1)2(k2+4(k+1))4                  =(k+1)2(k2+4k+4)4                  =(k+1)2((k+2)2)4                  =(k+1)2(k+1+1)2)4                  =(k+1)2((k+1)+1)2)4 Example2:Showthat1+3+5+…+(2n−1)=n2 Solution:  Step1:Resultistrueforn=1 Thatis1=(1)2 (True) Step2:Assumethatresultistrueforn=k 1+3+5+…+(2k−1)=k2  Step3: Checkforn=k+1 i.e.1+3+5+…+(2(k+1)−1)=(k+1)2  Wecanwritetheaboveequationas, 1+3+5+…+(2k−1)+(2(k+1)−1)=(k+1)2 Usingstep2result,weget k2 +(2(k+1)−1)=(k+1)2 k2 +2k+2−1=(k+1)2  k2 +2k+1=(k+1)2 (k+1)2 =(k+1)2  L.H.S.andR.H.S.aresame. Sotheresultistrueforn= k+1 Bymathematicalinduction,thestatementistrue. Weseethatthegivenstatementisalsotrueforn=k+1.Hencewecansaythatbytheprincipleofmathematicalinductionthisstatementisvalidforallnaturalnumbersn. Example3:  Showthat22n-1isdivisibleby3usingtheprinciplesofmathematicalinduction. Toprove:22n-1isdivisibleby3 AssumethatthegivenstatementbeP(k) Thus,thestatementcanbewrittenasP(k)=22n-1isdivisibleby3,foreverynaturalnumber Step1:Instep1,assumen=1,sothatthegivenstatementcanbewrittenas P(1)=22(1)-1= 4-1=3.So3isdivisibleby3.(i.e.3/3=1) Step2:Now,assumethatP(n)istrueforallthenaturalnumber,sayk Hence,thegivenstatementcanbewrittenas  P(k)= 22k-1isdivisibleby3. Itmeansthat22k-1=3a(whereabelongstonaturalnumber) Now,weneedtoprovethestatementistrueforn=k+1 Hence,  P(k+1)=22(k+1)-1  P(k+1)=22k+2-1 P(k+1)=22k.22–1 P(k+1)=(22k.4)-1 P(k+1)=3.2k+(22k-1) Theaboveexpressioncanbewrittenas P(k+1)=3.22k+3a Now,take3outside,weget P(k+1)=3(22k+a)=3b,where“b”belongstonaturalnumber Itisprovedthatp(k+1)holdstrue,wheneverthestatementP(k)istrue. Thus,22n-1isdivisibleby3isprovedusingtheprinciplesofmathematicalinduction RelatedLinks WholeNumbers RealNumbers RationalNumbers IrrationalNumbers Practiceproblems Provethat1×1!+2×2!+3×3!+…+n×n!=(n+1)!–1forallnaturalnumbersusingtheprinciplesofmathematicalinduction. Provethat4n–1isdivisibleby3usingtheprincipleofmathematicalinduction Usetheprinciplesofmathematicalinductiontoshowthat2+4+6+…+2n=n2+n,forallnaturalnumbers FrequentlyAskedQuestiononthePrincipleofMathematicalInductionWhatismeantbymathematicalinduction? Mathematicalinductionisdefinedasamethod,whichisusedtoestablishresultsforthenaturalnumbers.Generally,thismethodisusedtoprovethestatementortheoremistrueforallnaturalnumbers Writedownthetwostepsinvolvedintheprinciplesofmathematicalinduction? Thetwostepsinvolvedinprovingthestatementare: Provingthatthegivenstatementholdstruefortheinitialvalue.Thisiscalledthebasestep Instep2,provingthatthestatementistrueforthenthvalue,andalsoprovingthattrueforthe(n+1)thiterationalso.Thisstepiscalledtheinductionstep Whydoweusemathematicalinduction? Mathematicalinductionistypicallyusedtoprovethatthegivenstatementholdstrueforallthenaturalnumbers. Whatismeantbyweakandstronginduction? Inweakinduction,itisassumedthatonlyaparticularstatementholdstrueatthekthstep.Butinstronginduction,thegivenstatementholdstrueforallthestepsfrombasetothekthstep. Mentionthreedifferenttypesofmathematicalinduction Thedifferenttypesofmathematicalinductionare: Firstprincipleofmathematicalinduction Secondprincipleofmathematicalinduction Secondprincipleofmathematicalinduction(variation) ToknowmoreaboutmathvisitBYJU’S–TheLearningAppandlearnwitheasebywatchingtheinteractivevideos.WewishyouHappylearning! TestyourKnowledgeonPrincipleofMathematicalInduction Q5 PutyourunderstandingofthisconcepttotestbyansweringafewMCQs.Click‘StartQuiz’tobegin! Selectthecorrectanswerandclickonthe“Finish”buttonCheckyourscoreandanswersattheendofthequiz StartQuiz Congrats! VisitBYJU’SforallMathsrelatedqueriesandstudymaterials Yourresultisasbelow 0outof0arewrong 0outof0arecorrect 0outof0areUnattempted ViewQuizAnswersandAnalysis MATHSRelatedLinks RemainderTheorem LcmFormula MathematicsFormulaOfClass12 MedianOfGroupedData DifferenceBetweenRowsAndColumns SquareRootAndCubeRoot RomanNumber WhatIsALinearEquation Euclid'sPostulates ComplexNumbersClass11 LeaveaCommentCancelreply YourMobilenumberandEmailidwillnotbepublished.Requiredfieldsaremarked* * SendOTP DidnotreceiveOTP? 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