Question 122
Suppose an infinite sequence $\{a_n\}$ satisfies $a_0 = x$, $a_1 = y$, $a_{n+1} = \frac{a_n a_{n-1} + 1}{a_n + a_{n-1}}$, $n = 1, 2, \dots$.
(1) Find all real numbers $x$ and $y$ that satisfy the statement: there exists a positive integer $n_0$, such that, for $n \ge n_0$, $a_n$ is a constant.
(2) Find an explicit expression for $a_n$.
(1) Find all real numbers $x$ and $y$ that satisfy the statement: there exists a positive integer $n_0$, such that, for $n \ge n_0$, $a_n$ is a constant.
(2) Find an explicit expression for $a_n$.