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
 A: This is equivalent to:

Show for any $a$ where $a+a^{-1} \in \mathbb{Z}$ that $a^n+a^{-n} \in \mathbb{Z}$ for all $n\in\mathbb{Z}$.

If you need to prove a claim of the form

If $A$ then $B$

you start off with "assume $A$." From there you proceed to do whatever you need to do to arrive at $B$. In the context of your problem, all you have to do is say something along the lines of "suppose $a$ is such that $a+a^{-1} \in \mathbb{Z}$." If you are proving this with induction, that will complete the base case. Then in a similar fashion you state the induction hypothesis. "Suppose for all $n \in \{1,2,\ldots,k\}$ that $a^{n}+a^{-n} \in \mathbb{Z}$." Now proceed from here to do whatever you need to do to show $a^{k+1}+a^{-(k+1)} \in \mathbb{Z}$. 
A: 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).
$$(a+a^{-1})(a^n+a^{-n})-(a^{n-1}+a^{-(n-1)})=(a^{n+1}+a^{-(n+1)})\in\mathbb Z\ \ \ \square$$
