Sum of the series: $(1 + x) + (1 + x + x^{2}) + (1 + x + x^{2} + x^{3}) + . . .$ upto $n$ terms

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

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\frac{1}{1 - x} [n - \frac{x^{3} (1 - x^{n})}{1 - x}]

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\frac{1}{1 - x} [n - \frac{x (1 - x^{n})}{1 - x}]

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Solution

The correct option is A $\frac{1}{1 - x} [n - \frac{x^{2} (1 - x^{n})}{1 - x}]$
We have, $(1 + x) + (1 + x + x^{2}) + (1 + x + x^{2} + x^{3}) + . . .$ upto $n$ terms
$= \frac{1 - x^{2}}{1 - x} + \frac{1 - x^{3}}{1 - x} + \frac{1 - x^{4}}{1 - x} + . . .$ to $n$ terms
$= \frac{1}{1 - x} [(1 + 1 + 1 + 1 + . . . n t e r m s) - (x^{2} + x^{3} + x^{4} + . . . n t e r m s)]$
$= \frac{1}{1 - x} [n - \frac{x^{2} (1 - x^{n})}{1 - x}]$

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