In the series $a + (a + d) r + (a + 2 d) r^{2} + . . . . (a + ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ n - 1 . d) r^{n - 1}$ . Find the sum if $r < 1$ .

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Solution

Given series is $a + (a + d) r + (a + 2 d) r^{2} + . . . . (a + ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ n - 1 . d) r^{n - 1}$
Here $u_{n} = (a + ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ n - 1 . d) r^{n - 1}$
Consider,
$lim n \to \infty \frac{u_{n}}{u_{n - 1}}$
$= lim n \to \infty \frac{a + (n - 1) d}{a + (n - 2) d} . r = r$ ;
Thus if $r < 1$ the series is convergent, and the sum is finite.

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