Let `alpha = sum_(r = 0)^n (4r^2 + 2r + 1)^nC_r` and `beta = sum_(r =0)^n` `("^nC_r)/(r + 1) + 1/(n+1)`, if `140 < (2 alpha)/(beta) < 248` then find n.

Find sum_(r=0)^n(r+1)*"^nC_rx^r

Evaluate sum_(r=0)^n(r+1)^2*"^nC_r

Evaluate sum_(r = 0)^(n) 3^r ""^nC_r

Let a, b, c be the sides of Delta ABC opposite to angles A, b,C respecitvely. Let alpha = sum_(r=0)^(n) ""^(n)C_(r) b^(n-r) c^(r) cos{rB - (n-r)C} and beta = sum_(r=0)^(n) ""^(n)C_(r) b^(n-r) c^(r) sin{rB - (n-r)C} Statement -1: alpha = alpha^(n) Statement-2: beta = alpha^(n)

Evaluate sum_(r = 1)^(n) ""^nC_r 2^r

Prove that sum_(r=0)^(n)nC_(r)3^(r)=4^(n)

prove that sum_(r=0)^(n)3^(r)nC_(r)=4^(n)

sum_(r=0)^(n)nC_(r)(sin rx) is equal to

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