MEMO 2023 T8

Let $A$ and $B$ be positive integers. Consider a sequence of positive integers $(x_n)_{n \geq 1}$ such that

$$x_{n+1} = A \cdot \gcd(x_n, x_{n-1}) + B \quad \text{for every } n \geq 2.$$

Prove that the sequence attains only finitely many different values.

Remark. We denote by $\gcd(a, b)$ the greatest common divisor of positive integers $a$ and $b$.