Question 50
Let $a$, $b$ be two distinct odd positive integers. Define a sequence $(a_n)_{n \ge 1}$ by
$$
a_1 = a,\quad a_2 = b,\quad a_n \text{ is the largest odd divisor of } a_{n-1} + a_{n-2}, \text{ for all } n \ge 3.
$$
Show that there exists a natural number $N$ such that $a_n = \text{gcd}(a, b)$ for all $n \ge N$.
$$
a_1 = a,\quad a_2 = b,\quad a_n \text{ is the largest odd divisor of } a_{n-1} + a_{n-2}, \text{ for all } n \ge 3.
$$
Show that there exists a natural number $N$ such that $a_n = \text{gcd}(a, b)$ for all $n \ge N$.