How to prove a sum formula about powers of 2 and 3?

Question

I discovered this yesterday and I am wondering:

1. Is this a well known formula?
2. How do I go about proving something like this? (Beyond induction.)

$$2^{2n} = 3^n+\sum_{k=0}^{n-1}3^{n-k-1}2^{2k}$$

Thank you. Any thoughts on this would be greatly appreciated.

P.s: For notation purposes, when $n=0$ the sum is not evaluated.

Answer 1

It is: $$3^n+\sum_{k=0}^{n-1}3^{n-k-1}2^{2k}=3^n+3^{n-1}\sum_{k=0}^{n-1}\left(\frac43\right)^{k}=3^n+3^{n-1}\frac{\left(\frac43\right)^n-1}{\frac43-1}=\\ 3^n+(4^n-3^n)=4^n=2^{2n}.$$