Solve Recursive Function: A_{n+1}= (8/9)A_n

[wimma]

Homework Statement

Solve the recursive function:

A_{n+1} = (8/9)A_{n} + (24/9)*(20/9)^n

We want a closed formula for A_n

Homework Equations

I'm just doing this to work out the surface area of the Menger Sponge.

I know the formula's probably out there, but I get this recursive formula and I don't know how to solve it.

The Attempt at a Solution

I tried getting it in terms of A_{0}, which is 6, but I couldn't seem to simplify it... please help?!

 

[MathematicalPhysicist]

First solve the homogenous equation, i.e A_{n+1}=8/9 A_{n}, then write a solution as:
A{n}= A*(8/9)^n + B * (20/9)^n
plug this solution to the equation to find B, and then plug n=0 A{0} to find A.