How Can You Evaluate This Infinite Series with a Recurrence Relation?

[saubbie]

Homework Statement

The question is to evaluate the infinite series of the Sum[(((-1)^n)*a(n))/10^n], as n goes from zero to infinity, and a(n) is the recurrence relation a(n)=5a(n-1)-6a(n-2) where a(0)=0, and a(1)=1

Homework Equations

I found the explicit equation for a(n)=3^n - 2^n, but I can't find how that will help. It doesn't really simplify the sum that I can tell.

The Attempt at a Solution

I think that if I could find a generating function for the recurrence relation, then it would probably be a lot easier to relate the series to something that I already know, but I am not sure how to find the generating function. Any help is much appreciated. Thanks a lot.

 

[HallsofIvy]

Since you know that a_n= 3ⁿ- 2ⁿ and obvious thing to do is to put it in the sum- it certainly DOES simplify it!
The sum becomes
$$\sum_{n=}^{\infty}\frac{(-1)^n(3^n- 2^n)}{10^n}= \sum_{n=0}^\infty\left(\frac{-3}{10}\right)^n}-\sum_{n=0}^\infty\left(\frac{-2}{10}\right)^n$$
both of which are geometric series.