With usual notations prove that
`C_1+2.C_2x + 3.C_3x^2 + ……+2n.C_(2n) x^(2n - 1) = 2n (1 + x)^(2n -1)`

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

With usual notations prove that C_1/C_0 + 2. C_2/C_1 + 3.C_3/C_2 + ……+n.(C_n)/(C_(n-1)) = (n(n +1))/(2) Hence prove that (15C_1)/(15C_0) + 2.(15C_2)/(15C_1) + 3. (15C_3)/(15C_2) +……..+ 15. (15C_15)/(15C_14) = 120

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

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

With usual notation prove that 2^2.C_0 + 3^2.C_1 + 4^2.C_2 + ……+ (n+2)^2. C_n = (n^2 + 9n + 16) 2^(n-2)

With usual notations find the value of 3.C_1 + 5.C_2 + 7.C_3 + ……….+n terms

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

1.C_0 + 3.C_1+ 3^2.C_2+….+3^n.C_n =

If C_r denotes ""^nC_r then show that C_0 + (C_1)/(2) + (C_2)/(3) x^2 + ………..+ C_n. (x^n)/(n + 1) = ((1 + x)^(n+1) - 1)/((n + 1)x)

Prove that C_(0)+3.C_(1)+5.C_(2)+….+(2n+1).C_(n)=(2n+2).2^(n-1).