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.
Consider the sequence defined by , and
for all . Prove that all the terms of the sequence are positive integers.
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.
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.