If `ka n dn` 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`

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

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 :

If n is a positive integer and C_(k)=.^(n)C_(k) then find the value of sum_(k=1)^(n)k^(3)*((C_(k))/(C_(k-1)))^(2)

sum_(k =1)^(n) k(1 + 1/n)^(k -1) =

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

Suppose m and n are positive integers and let S=sum_(k=0)^(n)(-1)^(k)(1)/(k+m+1)(nC_(k)) and T=sum_(k=0)^(m)(-1)^(k)1(k+n+1)(mC_(k)) then S-T is equal to

The value of sum_(r=1)^(n+1)(sum_(k=1)^(n)C(k,r-1))=