Sequences of integers are a favourite of olympiad problem writers since such sequences involve several different mathematical concepts, including for example, algebraic techniques, recursive relations, divisibility and primality.
Problem:
Consider the sequence defined by
,
and
for all
. Prove that all the terms of the sequence are positive integers.
Solution:
There is no magic or sure shot or short cut formula to such problems. All I say is the more you read, the more rich your imagination, the more you try to solve on your own.
We have
Replacing n by yields,
and we obtain
This is equivalent to
or for all
. If n is even, we obtain
while if n is odd,
It follows that , if n is even,
and , if n is odd.
An inductive argument shows that all are positive integers.
More later,
Nalin Pithwa