Sum the following series:
$x + a, x^{2} + 2 a, x^{3} + 3 a . . .$ to $n$ terms.

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Solution

Let S be the sum of the series
Given series can be written as sum of two series whose sum is denoted by $S_{1}, S_{2}$
$S_{1} = x + x^{2} + x^{3} + . . . . . . . . . . . . x^{n}$
Clearly it is G.P. with common ratio $= x$
$S_{1} = \frac{x (x^{n} - 1)}{x - 1}$
$S_{2} = a + 2 a + 3 a + 4 a . . . . . . . . . . . . n a$
This series is an A.P with common diff. $= a$
$S_{2} = \frac{n}{2} \times (a + n a)$
$S = S_{1} + S_{2}$
$\Rightarrow S = \frac{x (x^{n} - 1)}{x - 1} + \frac{n a + n^{2} a}{2}$

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