Prove that `2^k(n,0)(n,k)-2^(k-1)(n,1)(n-1,k-1)+2^(k-2)(n,2)(n-2,k-2)-.....+(-1)^k(n,k)(n-k,0)=(n,k)`.

If n and k are positive integers, show that 2^k(nC 0)(n k)-2^(k-1)(nC1)(n-1Ck-1)+2^(k-2)(nC2)((n-2k-2))_dot-...+(-1)^k(nCk)+(n-kC0)=(nC k)w h e r e(n C k) stands for ^n C_kdot

2 ^ (k) ([n0]) ([nk]) - 2 ^ (k-1) ([n1]) ([n-1k-1]) + 2 ^ (k-2) ([n2]) ([nk-2]) - ,,, + (- 1) ^ (k) ([nk]) ([nk]) ([n0]) ([n0])

prove that sum_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))

Prove that : sum_(i=0)^r((n+i),(k))=((n+r+1),(k+1))-((n),(k+1))

If Delta_k=|(1,n,n),(2(k-1),n(n+1),n(n-1)),(4k(k^2-1),n(n+1)(n^2-4),n(n+2)(n^2-1))|, then f(n)=sum_(k=1)^n Delta_k+n is a polynomial of degree (i)0 (ii)1 (iii)6 (iv)7

sum_(k=1)^(oo)k(1-(1)/(n))^(k-1)=a*n(n-1)bn(n+1)c*n^(2)d.(n+1)^(2)

For the natural numbers n,k,<=tT(n,k)=(1)/((n+1)!)+(1)/((n+2)!)+(1)/((n+3)!)+......(1)/((n+k)!)

Prove that sum_(k=1)^(n-1) ""^(n)C_(k)[cos k x. cos (n+k)x+sin(n-k)x.sin(2n-k)x]=(2^(n)-2)cos nx .