Find a sequence such that $a_n^2=(a_{n-1}^2+a_{n+1}^2)/2$ for $n=2,3,\ldots$

Question

Suppose $a_n \in \mathbb{N}$ are natural numbers such that: $$a_n^2=\frac{a_{n-1}^2+a_{n+1}^2}{2}\quad (n=2,3,\ldots), \quad a_1=10 $$ Find $a_n$

Progress: So far I had come up with $a_n^2=100+(n-1)(a_2^2-100)$ result.

Answer 1

Hint : If $b_{n} = a_{n}^{2}$, then you have $$b_{n+1} - 2\,b_{n} + b_{n-1} = 0$$ which is a second order recurrence

Answer 2

I would suggest considering your equation as a recurrence relation and try substituting, $a_n=Ar^n$. Then you will get, $r=\pm 1$ and therefore you have the general solution as,

$$a_n=A+B(-1)^{n}$$

for $n\geq 2$.