`3. C_0 + 7. C_1 + 11. C_2 +…...+ (4n+3) . C_n=`

With usual notations prove that 2.C_0 + 7.C_1 + 12.C_2 + ……..+(5n + 2).C_n = (5n + 4).2^(n-1)

If C_o, C_1, C_2 ....,C_n C denote the binomial coefficients in the expansion of (1 + x)^n, then 1^3. C_1 + 2^3 . C_2 + 3^3 .C_3 + ... + n^3 .C_n, =

If C_(0),C_(1), C_(2),...,C_(n) denote the cefficients in the expansion of (1 + x)^(n) , then C_(0) + 3 .C_(1) + 5 . C_(2)+ ...+ (2n + 1) C_(n) = .

With usual notations prove that C_0 + 3.C_1 + 3^2.C_2 + ………..+3^n .C_n = 4^n

Prove that C_0.C_3 + C_1.C_4 + C_2.C_5 + …..+C_(n-3).C_n = ""^(2n)C_(n +3)

Prove that 3.C_0 + 6.C_1 + 12.C_2 + ………+3.2^n.C_n = 3^(n+1)

Prove that C_0 + 2.C_1 + 4.C_2 + 8.C_3 + ……+2^n.C_n = 3^n

If C_(0), C_(1), C_(2),..., C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then . 1. C_(1) - 2 . C_(2) + 3.C_(3) - 4. C_(4) + ...+ (-1)^(n-1) nC_(n)=

C_0 + 2.C_1 + 4.C_2 + …….+C_n.2^n = 243 , then n =

Prove that C_3 + 2.C_4+ 3.C_5 + ……..+ (n-2).C_n = (n-4).2^(n-1) + (n+2) where n > 3