AIME Problem

Consider the sequence defined by \(a_k = \frac{1}{k^2+k}\) for \(k \geq 1\).

Given that \(a_m + a_{m+1} + a_{m+2} + \ldots + a_{n-1} = \frac{1}{31}\), for positive integers \(m\) and \(n\) with \(m < n\). Find \(m+n\).

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