IN-TST 2009 Problem 3

Let $a,b$ be two distinct odd natural numbers.Define a Sequence ${ < a_n > }_{n\ge 0}$ like following: $a_1 = a a_2 = b a_n = \text{largest odd divisor of }(a_{n - 1} + a_{n - 2})$ . Prove that there exists a natural number $N$ such that $a_n = gcd(a,b) \forall n\ge N$ .