Suppose that $a_1 + a_2 + \dots + a_n =0$ for real numbers $a_1, a_2, \dots, a_n$ and $n \geq 2$. Set $a_{n+i}=a_i$ for $i=1, 2, \dots$. Prove that
$$\sum_{i=1}^n \frac{1}{a_i (a_i+a_{i+1}) (a_i+a_{i+1}+a_{i+2}) \dots (a_i+a_{i+1}+\dots+a_{i+n-2})} =0$$
if the denominators are nonzero.