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Show 1^(2).C(1)+2^(2).C(2)+3^(2). C(3)+…...

Show `1^(2).C_(1)+2^(2).C_(2)+3^(2). C_(3)+…+n^(2).C_(n)=n(n+1)2^(n-2)`

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1^(2)C_(1)-2^(2)C_(2)+3^(2)C_(3)-+(-1)^(n-1)n^(2)C_(n)=(1)(n^(2)*2^(n+1))/(n+1)(3)(2^(n+1))/(n-1)

If (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a+.^(n)C_(2)a^(2)+ . . +.^(n)C_(n)a^(n) , then prove that .^(n)C_(1)+2.^(n)C_(2)+3.^(n)3C_(3)+ . . .+n.^(n)C_(n)=n.2^(n-1) .

Knowledge Check

  • If C_(0), C_(1), C_(2),..., C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then . 1^(2). C_(1) - 2^(2) . C_(2)+ 3^(2). C_(3) -4^(2)C_(4) + ...+ (-1).""^(n-2)n^(2)C_(n)= .

    A
    0
    B
    `n^(2) 2^(n)`
    C
    `2^(n)`
    D
    `n(n-1) 2^(n-2)`

  • 1^(2). C_(1) - 2^(2) . C_(2)+ 3^(2). C_(3) -4^(2)C_(4) + ...+ (-1).""^(n-2)n^(2)C_(n)=

    A
    0
    B
    `2^(n)`
    C
    `n2^(n-1)`
    D
    `-n2^(n - 1)`

  • STATEMENT - 1 : If n is even, .^(2n)C_(1)+.^(2n)C_(3)+.^(2n)C_(5)+"….."+.^(2n)C_(n-1) = 2^(2n-1) . STATEMENT - 2 : .^(2n)C_(1) + .^(2n)C_(3)+ .^(2n)C_(5) + "……"+ .^(2n)C_(2n-1) = 2^(2n-1)

    A
    STATEMENT - 1 is true, STATEMENT - 2 is true and STATEMENT - 2 is correct explanation for STATEMENT - 1.
    B
    STATEMENT - 1 is true, STATEMENT - 2 is true and STATEMENT - 2 is not correct explanation for STATEMENT - 1.
    C
    STATEMENT-1 is true, STATEMENT-2 is false
    D
    STATEMENT-1 is false, STATEMENT-2 is true

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    If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)," prove that " + 3^(2) *C_(3) + …+ n^(2) *C_(n) 1^(2)*C_(1) + 2^(2) *C_(2) = n(n+1)* 2^(n-2) .

    If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(0) x^(n) , prove that (1*2) C_(2) + (2*3) C_(3) + …+ {(n-1)*n} C_(n) = n(n-1) 2^(n-2) .

    Prove that (.^(2n)C_(0))^(2) - (.^(2n)C_(1))^(2) + (.^(2n)C_(2))^(2) - …. + (.^(2n)C_(2n))^(2) = (-1)^(n) .^(2n)C_(n)

    Prove that C_(1)^(2) - 2. C_(2)^(2) + 3. C_(3)^(2) - …. -2n . C_(2n)^(2) = (-1)^(n). C_(n)

    C_(0)^(2)-C_(1)^(2)+C_(2)^(2) - …+ (-1)^(n)(C_(n))^(2)=0 or (-1)^(n//2) (n !)/((n//2)! (n//2)!) according as n is odd or even

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