Thread: How Good Are You At Reasoning?

I assume that by whole, one means integral. Otherwise, there's more work to do.

Originally Posted by Xei

2. x is any number on the number line such that x + 1/x is a whole number. Prove x^n + 1/x^n is also a whole number, for any positive whole number n.

Using strong induction, the one case is given to us. Specifically, x¹ + 1/x¹ is whole by the terms of the problem.

For the inductive case, note that (x^n-1 + 1/x^n-1)(x + 1/x) - (x^n-2 + 1/x^n-2) is whole as, by the inductive hypothesis, each term in brackets is whole. But trivial algebraic simplification reduces this to xⁿ + 1/xⁿ. Hence we've proven it for n as well.