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 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 :

sum_(k=m)^n kC_r

Let k=1^@ , then prove that sum_(n=0)^88 1/(cosnk* cos(n+1)k)=cosk/sin^2k

Find the sum_(k=1)^(oo) sum_(n=1)^(oo)k/(2^(n+k)) .

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.

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

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

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