Question 35
Consider the sequence $(a_n)_{n \in \mathbb{N}}$ where $a_1 = 2$ and for $n \ge 1$ we have
$$
a_{n+1} = \left(1 + \frac{1}{n}\right)^n a_n.
$$
Prove that there are infinitely many $n$ such that $\frac{1}{n+1}a_n a_{n-1} \dots a_1$ is a perfect square of a natural number.
$$
a_{n+1} = \left(1 + \frac{1}{n}\right)^n a_n.
$$
Prove that there are infinitely many $n$ such that $\frac{1}{n+1}a_n a_{n-1} \dots a_1$ is a perfect square of a natural number.