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Question: The sum of the series \(\frac{2}{1.2}\)+ \(\frac{5}{2.3}\). 2 + \(\frac{10}{3.4}\). 2<sup>2</sup>+ \...

The sum of the series 21.2\frac{2}{1.2}+ 52.3\frac{5}{2.3}. 2 + 103.4\frac{10}{3.4}. 22+ 174.5\frac{17}{4.5}. 23 + …… to n terms is-

A

nn+1\frac{n}{n + 1}2n+1

B

n+1n\frac{n + 1}{n}2n+1

C

nn+1\frac{n}{n + 1}2n

D

n+1n\frac{n + 1}{n}2n

Answer

nn+1\frac{n}{n + 1}2n

Explanation

Solution

The given series

= ∑r=1nr2+1r(r+1)\sum_{r = 1}^{n}\frac{r^{2} + 1}{r(r + 1)}2r–1 = ∑1n(2rr+1−r−1r)\sum_{1}^{n}\left( \frac{2r}{r + 1} - \frac{r - 1}{r} \right)2r–1

= ∑1n(rr+12r−r−1r2r−1)\sum_{1}^{n}\left( \frac{r}{r + 1}2^{r} - \frac{r - 1}{r}2^{r - 1} \right)= nn+1\frac{n}{n + 1}2n