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Principle of Mathematical Induction If it is known that  some statement is true for  Principle of Mathematical Induction If it is known that  some statement is true for

Principle of Mathematical Induction If it is known that some statement is true for - PDF document

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Principle of Mathematical Induction If it is known that some statement is true for - PPT Presentation

Prove that any positive integer n 1 is either a prime or can be represented as product of primes factors 2 Set contains all positive integers from 1 to 2 Prove that among any 1 numbers chosen from there are two numbers such that one is a factor of ID: 33072

Prove that any positive

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MathematicalInductioninAlgebra 1. Provethatanypositiveintegern�1iseitheraprimeorcanberepresentedasproductofprimesfactors. 2. SetScontainsallpositiveintegersfrom1to2n.Provethatamonganyn+1numberschosenfromStherearetwonumberssuchthatoneisafactoroftheother. 3. Provethatif(x+1=x)isintegerthen(xn+1=xn)isalsointegerforanypositiveintegern. 4. ForsequenceofFibonaccinumbersu1=1;u2=1;uk+1=uk+uk�1,k=2;3;:::provetheformulauk+m=uk�1um+ukum+1 5. Provethefollowingidentities: (a) 12+22+32++n2=n(n+1)(2n+1)=6 (b) 13+23+33++n3=n2(n+1)2=4 (c) 123+234++n(n+1)(n+2)=n(n+1)(n+2)(n+3)=4 (d) 11!+22!++nn!=(n+1)!�1 6. Provethefollowingdivisibilities: (a) 6=n3+5n (b) 7=(62n�1+1) (c) 3n+1=23n+1 7. Provethefollowinginequalities: (a) 1=(n+1)+1=(n+2)+1=(n+3)++1=2n�13=24(n�1) (b) 1=12+1=22+1=32++1=n22 (c) (x1++xn)=n(x1xn)1=n,wherex1;:::;xnarepositivenumbers 4. Pentagonalnumbers.Tode nesequenceofpentagonalnumbersPm(n)oneneedstopackballsintoregularm-gonswithnballsoneachside.Forexample,P4(1)=1,P4(2)=4,P4(3)=9,P4(4)=16,P4(5)=25.(Lookatpicturesbelow)                                                        EstablishtheformulaforPm(n),wherem3. 5. Establishandproveanestimateforminimalnumberofoperationstosolvethe\TheTowersofHanoi"puzzle(problem1,MathematicalInductioninProcesses)MathematicalInductioninProcesses 1. \TheTowersofHanoi"isapuzzlewith3nailsand7rings,allofdi erentsizes.Initiallyallringsareonthesamenailindecreasingorderfromthebottomtothetop.Theprocedureisremovingthetopringfromanynailandplacingitonanothernail.Itisnotallowedtoplaceabiggerringonthetopofasmallerone.Isitpossibletoreplacealltheringsfromonenailontoanotherbyapplyingthisprocedure?Whatistheminimalnumberofoperationsneededtosolvethepuzzle?Establishandprovetheformula. 2. Eachofnidenticalinshapejarsis lledwithapaintto(n�1)=nofitsvolume.Notwojarscontainthesamekindofthepaint.Itisallowedtopouranyamountofpaintfromonejartoanother.Couldonegetthesamemixtureinalljars?Paintisnotdisposableandthereisnootheremptyvessels. 3. Abandofnpiratesdivideapileoftreasuresbetweenthemselves.Eachwantstobesurethathegetsnolessthan1=n{thofthepile.Findafairwaytosplitthepilebetweenthem(sothatanyonecanblamenooneexceptmaybehimself).Thepirates'opinionsonsizeofthepilescouldbedi erent. 4. (TournamentofTowns2005,FallRound)Thereare1000potseachcontainingvaryingamountsofjam,notmorethan1/100-thofthetotal.Eachday,exactly100potsaretobechosen,andfromeachpot,thesameamountofjamiseaten.Provethatitispossibletoeatallthejamina nitenumberofdays.