Question

If $C_0,C_1,C_2,\dots ,C_n$ denote the coefficients in the expansion of ${(1+x)}^n$, then the value of $C_1+2C_2+3C_3+\cdots +n{C}_n$ is

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

Answer 1

Answer: (A)

$n\cdot 2^{n-1}$

Since,
${(1+x)}^n=C_0+x\cdot C_1+x^2\cdot C_2+\cdots +x^n\cdot C_n$
On differentiating both sides with respect to $x$, we get
$n{(1+x)}^{n-1}=C_1+2x{C}_2+\cdots +n{x}^{n-1}{C}_n$
Put $x=1$, we get
$n{\left(2\right)}^{n-1}=C_1+2\cdot C_2+3\cdot C_3+\cdots +n\cdot C_n$