Let
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be a positive integer. Show that if there exists a sequence
![a_0](/media/m/6/1/1/61182666f636aa197c5d27a9b3376b53.png)
,
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
, ... of integers satisfying the condition
![a_n=\frac{a_{n-1}+n^k}{n} \text{,} \qquad \forall n \geqslant 1 \text{,}](/media/m/c/d/5/cd5cbf6fadbfa9e349960d673fe73d6b.png)
then
![k-2](/media/m/7/6/c/76c61a2e732fcaf067b937a9324bffad.png)
is divisible by
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
.
Proposed by Turkey
%V0
Let $k$ be a positive integer. Show that if there exists a sequence $a_0$, $a_1$, ... of integers satisfying the condition $$a_n=\frac{a_{n-1}+n^k}{n} \text{,} \qquad \forall n \geqslant 1 \text{,}$$ then $k-2$ is divisible by $3$.
Proposed by Turkey