The sequence of integers u_{n} is bounded and satisfies u_{n} = (u_{n-1} + u_{n-2} + u_{n-3}u_{n-4})/(u_{n-1}u_{n-2} + u_{n-3} + u_{n-4}). Show that it is periodic for sufficiently large n.

**Solution**

This is almost trivial. The u_{n} are bounded and integral, so there are only *finitely* many possible values. Hence there are only finitely many possible values for the 4-tuples (u_{n-1}, u_{n-2}, u_{n-3}, u_{n-4}). So we must eventually get a repeat. Suppose the t-tuples are the same for n = N and n = N+M. Then the recurrence relation implies that they must also be the same for n = N+1 and n = N+M+1, and for n = N+2 and N+M+2 and so on. In other words the sequence is periodic (with period M) from this point onwards.

© John Scholes

jscholes@kalva.demon.co.uk

25 Jan 2002