Question

With usual notations, $C_1+2^2{C}_2+3^2{C}_3+.......=$

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

Answer 1

The correct option is C

n(n+1)2^n-2

$C_0+C_{1}x+C_0{x}^2+....+C_n{x}^n=(1+x{)}^n$ ....(1)

Diff. w.r.t. x,

$C_1+2C_{2}x+3C_3{x}^2+\cdots +n{C}_n{x}^{n-1}=n(1+x{)}^{n-1}$ .

Multiplying by x,

$C_{1}x+2C_2{x}^2+3C_3{x}^3+\cdots +n{C}_n{x}^n=nx(1+x{)}^{n-1}$

Diff. w.r.t. x, we get

$C_1+2^2{C}_{2}x+3^2{C}_3{x}^2+\cdots$

$=n(1+x{)}^{n-1}+n(n-1\left)x\right(1+x{)}^{n-2}$

Putting x=1,

$C_1+2^2{C}_2+3^2{C}_3+\cdots =n{.2}^{n-1}+n(n-1)2^{n-2}$

$=n(n+1){.2}^{n-2}$.