Prove that the number
is rational if and only if n is even.
(I think going one way if n is even then its clear that its rational, what about assuming that n is even then the number is rational? Is it inductive?)
Certaintly, the terms of the sequence are integers.
Furthermore, the solution to this recurrence relation is given by: .
If is even then
Consider the Fibonacci sequence it satifies
If is odd then .
Therefore for odd we have .