Question

If $C_0,C_1,C_2....C_n$ denote the binomial coefficients in the expansion of $(1+x{)}^n$ , then the value of $\sum _{r=0}^n(r+1)C_r$ is

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

Answer 1

The correct option is C $(n+2)2^{n-1}$

$\sum (r+1)C_r$
$=\sum \left(r\right)C_r+\sum C_r$
$={\frac{d(1+x{)}^n}{dx}}_{x=1}+2^n$
$=n(2{)}^{n-1}+2^n$
$=2^{n-1}(n+2)$