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Q.

If $s_{n}$ represents the sum of n terms of G.P. whose first term and common ratio are a and r respectively, then $s_{1} + s_{2} + s_{3} + . . . . + s_{n} =$

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a

$\frac{an}{1 - r} - \frac{ar (1 - r^{2 n})}{{(1 - r)}^{2} (1 + r)}$

b

$\frac{ar (1 - r^{2 n})}{(1 + r) {(1 - r)}^{2}}$

c

$\frac{an}{1 - r} - \frac{ar (1 - r^{n})}{{(1 - r)}^{2}}$

d

$\frac{ar (1 - r^{n})}{{(1 - r)}^{2}}$

answer is A.

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Detailed Solution

$\begin{array}{l} S_{n} = a (\frac{r^{n} - 1}{r - 1}) \\ S_{1} + S_{2} + \dots + S_{n} = \sum S_{n} \\ = \frac{a}{r - 1} \sum (r^{n} - 1) = \frac{a}{r - 1} ((\sum r^{n}) - \sum 1) \end{array} = \frac{a}{r - 1} [r [\frac{r^{n} - 1}{r - 1}] - n] = \frac{a}{{(r - 1)}^{2}} [r^{n + 1} - r - n r + 1]$

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If sn represents the sum of n terms of G.P. whose first term and common ratio are a and r respectively, then s1+s2+s3+....+sn=