If `n` and `k` are positive integers, show that `2^k( .^n C_0)(.^n C_k)-2^(k-1)(.^n C_1)(.^(n-1) C_k-1)+2^(k-2)(.^n C_2)((n-2k-2))_dot-...+ (-1)^k(^n C_k)+(.^(n-k) C_0)=(.^n C_k)w h e r e(.^n C_k)` stands for `.^n C_k.`

Show that sum_(k=m)^n ^kC_r=^(n+1)C_(r+1)-^mC_(r+1)

If n is a positive integer and C_(k)=""^(n)C_(k) , then the value of sum_(k=1)^(n)k^(3)((C_(k))/(C_(k-1)))^(2) is :

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

If k a n d n are positive integers and s_k=1^k+2^k+3^k++n^k , then prove that sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+1)dot

Show that lim_(nrarr infty) sum_(k=0)^(n)(""^(n)Ck)/(n^k(k+3))=e-2

sum_(r=1)^k(−3)^(r−1) .^(3n)C_(2r−1)=0, where k=(3n)/2 and n is an even integer

Evaluate : sum_(k=1)^n (2^k+3^(k-1))

Let n inN and k be an integer ge0 such that S_(k)(n)=1^(k)+2^(k)+3^(k)+ . . . +n^(k) Statement-1: S_(4)(n)=(n)/(30)(n+1)(2n+1)(3n^(2)+3n+1) Statement -2: .^(k+1)C_(1)S_(k)(n)+.^(k+1)C_(2)S_(k-1)(n)+ . . . +.^(k+1)C_(k)S_(1)(n)+.^(k+1)C_(k+1)S_(0)(n)=(n+1)^(k+1)-1

If C_(r) = ""^(n)C_(r) and (C_(0) + C_(1)) (C_(1) + C_(2)) … (C_(n-1) + C_(n)) = k ((n +1)^(n))/(n!) , then the value of k, is

If the value of (n + 2) . ""^(n)C_(0) *2^(n+1) - (n+1) * ""^(n)C_(1)*2^(n) + n* ""^(n)C_(2) * 2^(n-1) -... is equal to k(n +1) , the value of k is .