# A strange problem

Algebra Level 4

$\sum_{i=1}^{n} \dfrac{1}{a_{i} ( a_{i} + a_{i+1})( a_{i} + a_{i+1} + a_{i+2} ) \cdots (a_{i} + a_{i+1} + \cdots + a_{i+n-2}) }$

Given are the real numbers $$a_{1}, a_{2}, a_{3}, \ldots , a_{n}$$ whose sum is 0. And $$a_{n+m} = a_{m}$$ where $$m = 1,2,3,\ldots, n$$.

Determine the value of the sum above.

Assume all the denominators are non-zero.

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