Question

If $(1 + x)^n =C_0 + C_1x + C_2x^2 + ..... + C_n x^n$, then the value of $C_0 + 2C_1 + 3C_2 +....+ (n + 1) C_n$ will be

• A. $(n +2) 2^{n-1}$
• B. $(n +1) 2^{n}$
• C. $(n +1) 2^{n-1}$
• D. $(n +2) 2^{n}$

Answer 1

Answer: (A)

Since, $x(1 + x)^n$
$=xC_0 + C_1 x^2 +C_2x^3 + .... +C_n x^{n+1}$
On differentiating w.r.t. x, we get
$(1 +x)^n +nx (1+x)^{n-1}$
$ = C_0 + 2C_1x +3C_2x^2 +... +(n+1) C_nx^n$
Put x = 1, we get
$C_0 +2C_1 +3C_2 +... +(n +1) C_n$
$ = 2^n +n2^{n-1} = 2^{n -1} (n +2)$