find $a_n$, $a_n=\sqrt{\frac{a_{n-1}^2+a_{n+1}^2}{2}}, a_1=10(a_n\in \mathbb{N},n=2,3,4,\cdots)$ I would appreciate if somebody could help me with the following problem
Q: find $a_n$
$$a_n=\sqrt{\frac{a_{n-1}^2+a_{n+1}^2}{2}}, a_1=10(a_n\in \mathbb{N},n=2,3,4,\cdots)$$

Follow Ewan's answer. 
Obviously, $c_2\ge0$, otherwise $b_n=a_n^2=c_1+c_2n<0$ for large $n$.
If $c_2>0$, then $a_{n+1}-a_{n}>0$.
However, $a_{n+1}-a_{n}=\sqrt{c_1 + c_2 n+c_2}-\sqrt{c_1 + c_2 n}=\frac{c_2}{\sqrt{c_1 + c_2 n}+\sqrt{c_1 + c_2 n+c_2}}<\frac{c_2}{2\sqrt{c_1+c_2n}}$
So for $n>c_2-c_1/c_2$, $0<a_{n+1}-a_n<\frac{c_2}{2\sqrt{c_2^2}}=1/2$, which contradicts the condition that $a_n$ and $a_{n+1}$ are both natural number.
So $c_2=0$, and $c_1=a_1-0*1=10$,
In conclusion $a_n=c_1+0*n=a_1=10$.
