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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.

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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.

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1) $a_{1}=\frac{(5^1-1)}{4}=1$ - that's ok.

2) $a_{n}=\frac{5^n-1}{4}$

3)$a_{n+1}=5a_{n}+1=5\frac{(5^n-1)}{4}+1=\frac{5^{n+1}-5}{4}+\frac{4}{4}=\frac{5^{n+1}-1}{4}=a_{n+1}$

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