Question 4
Let $f_0,f_1,\ldots$ be the Fibonacci sequence: $f_0=f_1=1$, $f_n=f_{n-1}+f_{n-2}$ if $n\ge2$. 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 $$a\left(\frac1{f_1}+\frac1{f_1f_2}+\frac1{f_1f_2f_3}+\cdots+\frac1{f_1f_2\cdots f_n}\right)$$ is an integer.