Question

If $C_0,C_1,C_2....,C_n$ are Binomial Coefficients, such that $S_n=\sum _{r=0}^n\frac{1}{{C_r}^n}$ and $t_n=\sum _{r=0}^n\frac{r}{{C_r}^n}$ then $\frac{t_n}{s_n}$ equals

• A. $\frac{n}{2}$
• B. $\frac{n(n+1)}{2}$
• C. $\frac{n+1}{2}$
• D. None of these

Answer 1

The correct option is A $\frac{n}{2}$

$S_n=\sum _{r=0}^n\frac{1}{{}^n{C}_r}=\sum _{r=0}^{\frac{n}{2+1}}\frac{1}{{}^n{C}_r}+\frac{1}{{}^n{C}_{n-r}}$
$\Rightarrow S_n=\sum _{r=0}^{\frac{n+1}{2}}\frac{2}{{}^n{C}_r}=2\sum _{r=0}^{\frac{n+1}{2}}\frac{1}{{}^n{C}_r}$
$\therefore t_n=\sum _{r=0}^n\frac{r}{{}^n{C}_r}=\sum _{r=0}^{\frac{n+1}{2}}\frac{r+(n-r)}{{}^n{C}_r}$
$\Rightarrow n\sum _{r=0}^n\frac{1}{{}^n{C}_r}=n\left(\frac{S_n}{2}\right)=\frac{n}{2}{S}_n$

Thus $\frac{t_n}{S_n}=\frac{n}{2}$