The sequence of integers un is bounded and satisfies un = (un-1 + un-2 + un-3un-4)/(un-1un-2 + un-3 + un-4). Show that it is periodic for sufficiently large n.
Solution
This is almost trivial. The un are bounded and integral, so there are only finitely many possible values. Hence there are only finitely many possible values for the 4-tuples (un-1, un-2, un-3, un-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