Question

The sum of the series $\frac{2}{1 \cdot 2}+ \, \frac{5}{2 \cdot 3}\cdot 2 \, + \, \frac{10}{3 \cdot 4}\cdot 2^{2} \, + \, \frac{17}{4 \cdot 5}\cdot 2^{3} \, +$ ...... to $n$ terms is

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

Answer 1

Answer: (C)

The required sum
$=\displaystyle \sum _{r = 1}^{n}\frac{r^{2} + 1}{r \left(r + 1\right)}2^{r - 1}=\displaystyle \sum _{r = 1}^{n}\left(\frac{2 r}{r + 1} - \frac{r - 1}{r}\right)2^{r - 1}$
$= \, \displaystyle \sum _{r = 1}^{n}\left(\frac{r}{r + 1} 2^{r} - \frac{r - 1}{r} 2^{r - 1}\right)= \, \frac{n}{n + 1}\cdot 2^{n}$