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
On
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 $a^{k+1}+a^{-(k+1)} \in \mathbb{Z}$.
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).
$$(a+a^{-1})(a^n+a^{-n})-(a^{n-1}+a^{-(n-1)})=(a^{n+1}+a^{-(n+1)})\in\mathbb Z\ \ \ \square$$