A Long Sum

Algebra Level 3

The sum \[ \frac{3}{1^2 2^2} + \frac{5}{2^2 3^2} + \frac{7}{3^2 4^2} + \cdots + \frac{41}{20^2 21^2} \] can be written as \( \dfrac{a}{b} \) where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

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