Induction question with 'if' statement I have an induction homework question that I got stuck in the middle.
Prove by induction that if $a + a^{-1} \in \Bbb{Z}$ then for each $n \in \Bbb{N}$ the following is true:
$$a^{n} + a^{-n} \in \Bbb{Z}$$
If possible, I would like to understand the method for solving those kind of questions (I know induction, but the "if" part at the begining confuses me)
Thanks in advance,
Ron

So we have $a^1+a^{-1}\in\mathbb Z$ and $a^2+a^{-2}=(a^1+a^{-1})^2-2\in\mathbb Z$.  
Suppose $a^k+a^{-k}\in\mathbb Z$ for all $k\in\mathbb Z^+, k\le n$, where $a^n+a^{-n}\in\mathbb Z$.  
We will prove that then $a^{n+1}+a^{-(n+1)}\in\mathbb Z$ (so your statement follows by strong induction).
$$(a+a^{-1})(a^n+a^{-n})-(a^{n-1}+a^{-(n-1)})=(a^{n+1}+a^{-(n+1)})\in\mathbb Z\ \ \ \square$$
