Find the sum of the following arithmetico geometric sequence
$2, 5 x, 8 x^{2}, 11 x^{3} . . . . . .$ upto $n - t e r m s$

\frac{2}{1 - x} + \frac{3 x (1 - x^{n - 1})}{(1 - x)^{2}} - \frac{(3 n - 1) x^{n}}{1 - x}

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\frac{2}{1 - x} + \frac{3 x (1 - x^{n - 1})}{(1 - x)} + \frac{3 n (1 - x^{n})}{1 - x}

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\frac{1}{1 - x} + \frac{3 (1 - x^{n - 1})}{(1 - x)^{2}} + \frac{3 n (1 - x^{n})}{1 - x}

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\frac{2}{1 - x} + \frac{x (1 - x^{n - 1})}{(1 - x)^{2}} + \frac{n (1 - x^{n})}{1 - x}

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Solution

The correct option is B $\frac{2}{1 - x} + \frac{3 x (1 - x^{n - 1})}{(1 - x)^{2}} - \frac{(3 n - 1) x^{n}}{1 - x}$
Let $S = 2 + 5 x + 8 x^{2} + 11 x^{3} + . . . . + (3 n - 1) x^{n - 1}$ .....(1)
$\Rightarrow x S = 2 x + 5 x^{2} + 8 x^{3} + 11 x^{4} + . . . . + (3 n - 1) x^{n}$ .....(2)
$∴ e q (1) - e q (2)$
$\Rightarrow (1 - x) S = 2 + 3 x + 3 x^{2} + 3 x^{3} + . . . . + 3 x^{n - 1} - (3 n - 1) x^{n}$
$\Rightarrow (1 - x) S = 2 + 3. \frac{x (x^{n - 1} - 1)}{x - 1} - (3 n - 1) x^{n}$
$\Rightarrow S = \frac{2}{1 - x} + \frac{3 x (1 - x^{n - 1})}{(1 - x)^{2}} - \frac{(3 n - 1) x^{n}}{1 - x}$

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