The sum of first n terms of the series 35+2125+117125+ … is

35+2125+117125+........ =1-25+1-252+1-253+......... =n-25 1-25n1-25 =1-25 1-25n 53 =n-23+2n+13×5n

The sum of first n terms of the series 35+2125+117125+ … is

35+2125+117125+........ =1-25+1-252+1-253+......... =n-25 1-25n1-25 =1-25 1-25n 53 =n-23+2n+13×5n

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The sum of first $n$ terms of the series $\frac{3}{5} + \frac{21}{25} + \frac{117}{125} + \dots$ is

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$n + \frac{2^{n + 1}}{3 \times 5^{n}} - \frac{2}{3}$

$n - \frac{2^{n + 1}}{3 \times 5^{n}} - \frac{2}{3}$

$n + \frac{2^{n + 1}}{3 \times 5^{n}} + \frac{2}{3}$

$n - \frac{2^{n + 1}}{3 \times 5^{n}} + \frac{2}{3}$

answer is A.

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Detailed Solution

$\begin{array}{rcl} \frac{3}{5} + \frac{21}{25} + \frac{117}{125} + . . . . . . . . \\ = & 1 - \frac{2}{5} + 1 - {(\frac{2}{5})}^{2} + 1 - {(\frac{2}{5})}^{3} + . . . . . . . . . \\ = & n - \frac{2}{5} \frac{(1 - {(\frac{2}{5})}^{n})}{1 - \frac{2}{5}} \\ = & 1 - \frac{2}{5} (1 - {(\frac{2}{5})}^{n}) \frac{5}{3} \\ = & n - \frac{2}{3} + \frac{2^{n + 1}}{3 \times 5^{n}} \end{array}$