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

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