If Cr Means NCr Then C0/1+C2/3+C4/5+⋯ = - Byju's

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wiz-iconMyQuestionIconMyQuestionIcon1You visited us 1 times! Enjoying our articles? Unlock Full Access!Byju's AnswerStandard XIIMathematicsIntegration to Solve Modified Sum of Binomial CoefficientsIf Cr means n...Question

If Cr means nCr then C01+C23+C45+⋯=

A

2(n+1)n+1

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2nn

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2nn+1

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2(n+2)n+1

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The correct option is C

2nn+1

Consider the expansion (1+x)n=C0+C1x+C2x2+C3x3+C4x4+⋯+Cnxn . . .(i) Integrating both sides of (i) within limits - 1 to 1, we get ∫1−1(1+x)ndx=∫1−1(C0+C1x+C2x2+C3x3+C4x4+⋯+Cnxn)dx=∫1−1(C0+C2x2+C4x4+⋯)dx+∫1−1(C1x+C3x3+⋯)dx=2∫10(C0+C2x2+C4x4+⋯)dx+0 (By Prop. of definite integral) (since second integral contains odd function) (1+x)n+1n+1]1−1==2(C0x+C2x33+C4x55+⋯)]102n+1n+1=2(C0+C23+C45+⋯) Hence C0+C23+C45+⋯=2nn+1

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If Cr means nCr then C01+C23+C45+⋯=

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Q. If Cr=nCr and (C0+C1)(C1+C2)...(Cn−1+Cn)=k (C0C1C2...Cn) then the value of k equalsQ. C01+C23+C45+C67.....=View MoreJoin BYJU'S Learning ProgramGrade/Exam1st Grade2nd Grade3rd Grade4th Grade5th Grade6th grade7th grade8th Grade9th Grade10th Grade11th Grade12th GradeSubmitsimilar_iconRelated VideosthumbnaillockSum of Product of Binomial CoefficientsMATHEMATICSWatch in Appexplore_more_iconExplore moreIntegration to Solve Modified Sum of Binomial CoefficientsStandard XII MathematicsJoin BYJU'S Learning ProgramGrade/Exam1st Grade2nd Grade3rd Grade4th Grade5th Grade6th grade7th grade8th Grade9th Grade10th Grade11th Grade12th GradeSubmitbook_imageAI Tutorbook_imageTextbooksbook_imageQuestion Papersbook_imageInstall appCrossIcon

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