Geometric Series Variant

Question

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,

Answer 1

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.