If `p+q=1,` then show that `sum_(r=0)^n r^2^n C_rp^r q^(n-r)=n p q+n^2p^2dot`

Prove that sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot

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

Prove that sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot

If ""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q =1 sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = .

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

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

Let p_1,p_2,...,p_n and x be distinct real number such that (sum_(r=1)^(n-1)p_r^2)x^2+2(sum_(r=1)^(n-1)p_r p_(r+1))x+sum_(r=2)^n p_r^2 lt=0 then p_1,p_2,...,p_n are in G.P. and when a_1^2+a_2^2+a_3^2+...+a_n^2=0,a_1=a_2=a_3=...=a_n=0 Statement 2 : If p_2/p_1=p_3/p_2=....=p_n/p_(n-1), then p_1,p_2,...,p_n are in G.P.

If n in N, then sum_(r=0)^(n) (-1)^(n) (""^(n)C_(r))/(""^(r+2)C_(r)) is equal to .

The value of sum_(r=1)^(n)(""^(n)P_(r))/(r!) is

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