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

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))

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + c_(3) x^(3) + …+ C_(n) x^(n) , show that sum_(r=0)^(n) (C_(r) 3^(r+4))/((r+1)(r+2)(r+3)(r+4)) = (1)/((n+1)(n+2)(n+3)(n+4))(4^(n+4) -sum_(t=0)^(3) ""^(n+4)C_(t)) .

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

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

sum_(r = 0)^(n-1) (C_r)/(C_r + C_(r+1)) =

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

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

Statement-1 sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = nx (1 - x)^(n -1) Statement-2: sum_(r=0)^(n)r ""^(n)C_(r) x^(r) (-1)^(r) =0

Find the sum of sum_(r=1)^n(r^n C_r)/(^n C_(r-1) .