Question 132
Let $k$ be a positive integer. Show that if there exists a sequence $a_0, a_1, \dots$ of integers satisfying the condition
$$
a_n = \frac{a_{n-1} + n^k}{n} \text{ for all } n \ge 1,
$$
then $k-2$ is divisible by 3.
$$
a_n = \frac{a_{n-1} + n^k}{n} \text{ for all } n \ge 1,
$$
then $k-2$ is divisible by 3.