Question 29
Let $k$ be a positive integer. Show that if there exists a sequence $a_{0}, a_{1}, \ldots$ of integers satisfying the condition
$$
a_{n}=\frac{a_{n-1}+n^{k}}{n} \quad \text { for all } n \geq 1
$$
then $k-2$ is divisible by 3 .
$$
a_{n}=\frac{a_{n-1}+n^{k}}{n} \quad \text { for all } n \geq 1
$$
then $k-2$ is divisible by 3 .