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I want to find the expression for the following series. It is similar to a geometric series but I don't get how to find the answer.

$\sum^n_{i=1}a^{i-1}b^i$

thanks,

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It is a geometric series: the first term is $b$, the last term is $a^{n-1}b^n$, and the ratio is $ab$. –  Brian M. Scott Apr 9 '12 at 7:31

1 Answer 1

up vote 1 down vote accepted

You have

$$ \sum_{i=1}^n a^{i-1} b^i = b \left ( \sum_{i=1}^n a^{i-1} b^{i-1} \right ) $$

Now compute

$$\sum_{i=1}^n a^{i-1} b^{i-1} = \sum_{i=0}^{n-1} a^{i} b^{i} = \sum_{i=0}^{n-1} (ab)^{i} = \sum_{i=0}^{n-1} q^{i}$$

Hope this helps.

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this solved my issue, thanks. –  RushK Apr 9 '12 at 8:19

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