`C_0^2-C_1^2+C_2^2-C_3^2+.....+(-1)^nC_n^2=` (A) zero always (B) zero when n is even (C) zero when n is odd (D) non zero

C_0 - [C_1 -2.C_2+ 3.C_3-……..+(-1)^(n-1).n.C_n] =

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

Show that C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+.........+(-1)^(n)C_(n)^(2)=0 or (-1)^((n)/(2))C_((n)/(2)) according as n is odd or even.

C_1/C_0 + (2C_2)/C_1 + (3C_3)/C_2 + .... (nC_n)/C_(n-1) is

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3)x^(3) + …+ C_(n) x^(n) , then C_(0) - (C_(0) - C_(1)) + (C_(0) + C_(1) + C_(2))- (C_(0) + C_(1) + C_(2)+ C_(3)) + ...+ (-1)^(n-1) (C_0) + C_(1) + C_(2) + ...+ C_(n-1)) , when n is even integer is

1.C_0 + 3.C_1+ 3^2.C_2+….+3^n.C_n =