Question 11
Let $N$ be a positive integer. Define a sequence $a_{n}$, $n \geq 0$, by
$$
a_{0}=0, \quad a_{1}=1, \quad a_{n+1}+a_{n-1}=a_{n}\left(2-\frac{1}{N}\right) \quad(n \geq 1)
$$
Prove that $a_{n}\lt \sqrt{N+1}$ for all $n \geq 0$.
$$
a_{0}=0, \quad a_{1}=1, \quad a_{n+1}+a_{n-1}=a_{n}\left(2-\frac{1}{N}\right) \quad(n \geq 1)
$$
Prove that $a_{n}\lt \sqrt{N+1}$ for all $n \geq 0$.