Sum to $n$ terms of the series
$l + (1 + x) + (1 + x + x^{2}) + (1 + x + x^{2} + x^{3}) + \dots$

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

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

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

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

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Solution

The correct option is A $\frac{n}{1 - x} - \frac{x (1 - x^{n})}{(1 - x)^{2}}$
$r t h$ term of the series can be written as $t_{r} = 1 + x + x^{2} + x^{3} + . . . + x^{r - 1} = \frac{1 - x^{r}}{1 - x}$
$S_{n} = n \sum r = 1 t_{r} = n \sum r = 1 \frac{1 - x^{r}}{1 - x} = \frac{1}{1 - x} [(1 - x) + (1 - x^{2}) + (1 - x^{3}) + . . . + (1 - x^{n})] = \frac{1}{1 - x} [n - (x + x^{2} + x^{3} + . . . + x^{n})] = \frac{1}{1 - x} [n - \frac{x (1 - x^{n})}{1 - x}]$

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