Question 125
Let $f_0, f_1, \dots$ be the Fibonacci sequence: $f_0 = f_1 = 1$, $f_n = f_{n-1} + f_{n-2}$ if $n \ge 2$. Determine all possible positive integers $n$ so that there is a positive integer $a$ such that $f_n \le a \le f_{n+1}$ and that
$$
a\left(\frac{1}{f_1} + \frac{1}{f_1f_2} + \frac{1}{f_1f_2f_3} + \dots + \frac{1}{f_1f_2\dots f_n}\right)
$$
is an integer.
$$
a\left(\frac{1}{f_1} + \frac{1}{f_1f_2} + \frac{1}{f_1f_2f_3} + \dots + \frac{1}{f_1f_2\dots f_n}\right)
$$
is an integer.