A Long Sum

Algebra Level 3

The sum 31222+52232+73242++41202212 \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 ab \dfrac{a}{b} where aa and bb are positive coprime integers. Find a+ba + b.

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