With usual notations prove that
`2.C_0 + 7.C_1 + 12.C_2 + ……..+(5n + 2).C_n = (5n + 4).2^(n-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 C_1+2.C_2x + 3.C_3x^2 + ……+2n.C_(2n) x^(2n - 1) = 2n (1 + x)^(2n -1)

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

3. C_0 + 7. C_1 + 11. C_2 +…...+ (4n+3) . C_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

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

Prove that following i) 2.C_(0)+5.C_(1)+8.C_(2)+…..+(3n+2)C_(n)=(3n+4).2^(n-1)

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

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