Question 47
Let $a_1, a_2, a_3, \dots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity
$$
\frac{a_n + a_{n+1} + \dots + a_{n+k-1}}{k}
$$
is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square).
$$
\frac{a_n + a_{n+1} + \dots + a_{n+k-1}}{k}
$$
is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square).