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 2.C_0 + 7.C_1 + 12.C_2 + ……..+(5n + 2).C_n = (5n + 4).2^(n-1)

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

Find 1.C_0 + 3.C_1+ 3^2.C_2+….+3^n.C_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

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

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

3. C_0 + 7. C_1 + 11. C_2 +…...+ (4n+3) . C_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 : C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0