Question 173
Given a sequence $\{a_n\}$ of numbers satisfying $a_0 = 1$, $a_{n+1} = \frac{7a_n + \sqrt{45a_n^2 - 36}}{2}$, $n \in \mathbb{N}$.
Prove that
(1) for each $n \in \mathbb{N}$, $a_n$ is a positive integer.
(2) for each $n \in \mathbb{N}$, $a_n a_{n+1} - 1$ is a perfect square.
Prove that
(1) for each $n \in \mathbb{N}$, $a_n$ is a positive integer.
(2) for each $n \in \mathbb{N}$, $a_n a_{n+1} - 1$ is a perfect square.