$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$. Show that if $a_{1}=5$, the $16 \mid a_{81}$.
Source: China North MO
Tags: induction, number theory unsolved, number theory
$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$. Show that if $a_{1}=5$, the $16 \mid a_{81}$.