Deduce that: `sum_(r=0)^n .^nC_r (-1)^r 1/((r+1)(r+2)) = 1/(n+2)`

If x+y=1, prove that sum_(r=0)^n .^nC_r x^r y^(n-r) = 1 .

Prove that sum_(r = 1)^n r^3 ((n_C_r)/(C_(r - 1)))^2 = (n (n + 1)^2 (n+2))/(12)

Prove that sum_(r = 1)^(n+1) (2^(r +1) C_(r - 1) )/(r (r + 1)) = (3^(n+2) - 2n - 5)/((n+1)(n+2))

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

sum_(r=1)^(n) (-1)^(r-1) ""^nC_r(a - r) =

sum_(r=0)^n (-1)^r .^nC_r (1+rln10)/(1+ln10^n)^r

Assertion: If n is an even positive integer n then sum_(r=0)^n ("^nC_r)/(r+1) = (2^(n+1)-1)/(n+1) , Reason : sum_(r=0)^n ("^nC_r)/(r+1) x^r = ((1+x)^(n+1)-1)/(n+1) (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not te correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Find the sum sum_(r=1)^n r^n (^nC_r)/(^nC_(r-1)) .

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}

Prove that sum_(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)^n C_r=1/n .