Question

${\sum }_{r=0}^{n-1}\frac{{{}^{n}C}_r}{{{}^{n}C}_r+{{}^{n}C}_{r+1}}$

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

Answer 1

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

The given summation can be re-written as
${\sum }_{r=0}^{n-1}\frac{{}^n{C}_r}{{}^{n+1}{C}_{r+1}}$ ...(Using the properties of binomial coefficient)
$\to {\sum }_{r=0}^{n-1}\frac{r+1}{n+1}$
$=\frac{1}{n+1}+\frac{2}{n+1}+\frac{3}{n+1}+...\frac{n}{n+1}$
$=\frac{n(n+1)}{2(n+1)}$
$=\frac{n}{2}$