Question

If $C_0.C_1,C_2,...,C_n$ are the coefficients of the expansion of $(1+x{)}^n$, then the value of $\sum _0^n\frac{C_k}{k+1}$ is

• A. $0$
• B. $\frac{2^n-1}{n}$
• C. $\frac{2^{n+1}-1}{n+1}$
• D. None of these

Answer 1

Answer: (C)

$\frac{2^{n+1}-1}{n+1}$

$t_{r+1}=\frac{{{}^{n}C}_r}{r+1}=\frac{{{}^{n+1}C}_{r+1}}{n+1}$

Substitute $r=0,1,2,3,...n$, we get

$\sum _o^n\frac{C_k}{k+1}=\frac{1}{n+1}[{{}^{n+1}C}_1+{{}^{n+1}C}_2+{{}^{n+1}C}_3+....+{{}^{n+1}C}_{n+1}]=\frac{1}{n+1}[2^{n+1}-{{}^{n+1}C}_0]=\frac{2^{n+1}-1}{n+1}$