Find sum of series $n + (n - 1) x + (n - 2) x^{2} + . . . . + 2 x^{n - 2} + x^{n - 1}$

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Solution

consider,
$S = n + (n - 1) x + (n - 2) x^{2} + . . . . . . + x^{n - 1}$
$x S = n x + (n - 1) x^{2} + . . . . . + 2 x^{n - 1} + x^{n}$
$\Rightarrow x S - S = n + (n - (n - 1)) x$
$+ ((n - 1) - (n - 1)) x$
$+ (n - 2) - (n - 3) x^{3}$
$x^{n - 1} + x^{n}$
$\Rightarrow (x - 1) S = n + (x + x^{2} + . . . . . . x^{n - 1} + x^{n})$
$\Rightarrow (x - 17 S) = n + x (\frac{x^{n} - 1}{x - 1})$
$\Rightarrow X = \frac{n}{x - 1} + \frac{x (x^{n} - 1)}{(x - 1)^{2}}$

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