- #1

CynicusRex

Gold Member

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## Homework Statement

Algebra I.M. Gelfand, problem 134.

You know that x+1/x is an integer. Prove that x

^{n}+1/x

^{n}is an integer for any n = 1, 2, 3, etc.

## Homework Equations

## The Attempt at a Solution

I don't fully understand how the induction below proves anything. (Source)

I did attempt the solution, yet kept trying because I ignored similar answers to the one below, thinking I didn't prove anything.

The base case of n = 1 is true, and suppose it holds for all k < n in order to do the induction step. Then

$$(x^{n-1}+1/x^{n-1})(x+1/x)=x^n+1/x^{n-2}+x^{n-2}+1/x^n=(x^n+1/x^n)+(x^{n-2}+1/x^{n-2})$$

so

$$x^n+1/x^n=(x^{n-1}+1/x^{n-1})(x+1/x)-(x^{n-2}+1/x^{n-2})$$

which is an integer, so the result follows by induction.

I don't get this part: "

**suppose it holds for all k < n**"

Isn't that what you are supposed to prove?