Question

Let x be a real number such that x+1/x is an integer. Prove that x^n+1/x^n is an integer, for all positive integers n.
I attempted to prove this using strong induction and factored x^(k+1)+1/x^(k+1) but realized that for this to work, k+1 must be odd.
Do I divide into cases where the exponents are odd and even?