Question

The sum of the series
$\frac{2}{1\cdot2}+\frac{5}{2\cdot3}\cdot2+\frac{10}{3\cdot4}\cdot2^{2} +\frac{17}{4\cdot5}\cdot2^{3} +..... $ to $n$ terms is

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

Answer 1

Answer: (C)

The given series is
$\sum\limits_{r=1}^{n} \frac{r^{2}+1}{r\left(r+1\right)} 2^{r-1} = \sum\limits_{r=1}^{n} \left(\frac{2r}{r+1} - \frac{r-1}{r}\right)2^{r-1}$
$=\sum\limits_{r=1}^{n} \left(\frac{r}{r+1}2^{r} -\frac{r-1}{r}2^{r-1}\right) = \frac{n}{ n+1} 2^{n}$