How to simplify the product of $(x^{2^k}+x^{-2^k})$ over $k$?

Question

Simplify $$\left(x+\frac1x\right)\left(x^2+\frac1{x^2}\right)\left(x^4+\frac1{x^4}\right)\ldots\left(x^{2^{n-1}}+\frac1{x^{2^{n-1}}}\right)$$ for $n\in\Bbb N$.

Should I be using mathematical induction on $n$ for this, or is there another way?

Answer 1

am I gonna solve it by using mathematical induction or..

I don't know how you're going to solve it, or what comes after the ellipsis. But let me tell you, I multiplied that thing by $\displaystyle \left( x - \frac{1}{x}\right)$ and the craziest thing happened...

Answer 2

You will eventually use mathematical induction in your solution.

However, you have some work to do before you get to that point: the first thing you need to do before you can solve this problem is to figure out what's going on.

One of the usual approaches to solving problems where you don't know what's going on is to look at special cases. This is especially true in problems where you have something going on "$n$ times" or the like.

Before trying to tackle the general problem, you first tackle the $n=1$ case in isolation. Then you look at the $n=2$ case. Then the $n=3$ case, and maybe a few more.

Frequently, after working through these special cases, the it becomes clear how th general case is going to look. At that point, you write down what you think the answer is going to be, and then use some means (e.g. induction, as you thought) to verify that your answer is correct.