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If Cr denotes ""^nCr then show that ...

If `C_r` denotes `""^nC_r` then show that
`C_0 + (C_1)/(2) + (C_2)/(3) x^2 + ………..+ C_n. (x^n)/(n + 1) = ((1 + x)^(n+1) - 1)/((n + 1)x)`

Answer

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Knowledge Check

  • C_0 + (C_1)/(2) (4) + (C_2)/(3) (16) + …………..+(C_n)/(n + 1) (2^(2n))

    A
    `(5^(n+1) + 1)/(n-1)`
    B
    `(5^(n+1) - 1)/(4(n+ 1))`
    C
    `(5^(n+1) + 1)/(4(n + 1))`
    D
    `(5^(n+1) + 1)/(4(n -1))`

  • k. C_0 + k^2 . (C_1)/(2)+k^3. (C_2)/(3)+…..+ k^(n+1). (C_n)/(n+1)=

    A
    `((k+1)^(n+1)-1)/(n+1)`
    B
    `((k-1)^(n+1)-1)/(n+1)`
    C
    `((k-1))^(n+1)+1)/(n-1)`
    D
    `((k+1)^(n+1)+1)/(n-1)`

  • C_0+(C_1 x)/(2)+(C_2 x^2)/(3)+…...+(C_n x^n)/(n+1)=

    A
    `(1)/((n+1)x)`
    B
    `((1+x)^n)/((n+1)x)`
    C
    `((1+x)^(n+1))/((n+1)x)`
    D
    `((1+x)^(n+1)-1)/((n+1) x)`

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    Prove that : If n is a positive integer, then prove that C_(0)+(C_(1))/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1).

    2. C_0+ 2^2 (C_1)/(2)+2^3. (C_2)/(3)+…....+2^(n+1). (C_n)/(n+1)=

    C_0-(C_1)/(2)+(C_2)/(3)-…...+(-1)^n (C_n)/(n+1)=

    A : C_0-(C_1)/(2)+(C_2)/(3)+(C_2)/(3)-….+ (-1)^n (C_n)/(n+1)=(1)/(n+1) R : C_0 + (C_1)/(2)x. + (C_2)/(3) x^2 + (C_3)/(4) x^3 + ….+ (C_n)/ (n+1) .x^n = ((1+x)^(n+1)-1)/((n+1)x)

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