Proof by induction for a recursive sequence (closed form formula given) I was given the following:
A sequence is defined recursively by a0 = 0, and, for n>=1, an = 5an-1 + 1. Use induction to prove the closed form formula for an is an = (5n - 1) / 4.
So far for my proof, all I have is this:
an+1 = 5an + 1
=5 ((5n - 1) / 4) + 1
What do I do next? I've forgotten what I'm even trying to prove.

base case: $a_1=\frac{5^1-1}{4}=\frac{5-1}{4}=\frac{4}{4}=1$
Assume $a_n=\frac{5^n-1}{4}$
then $a_{n+1}=5a_n+1=5\frac{5^n-1}{4}+1=\frac{5^{n+1}-5}{4}+1=\frac{5^{n+1}-5+4}{4}=\frac{5^{n+1}-1}{4}$
as was desired.
