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Algebra Level 4

17.8.9+18.9.10+19.10.11..............+120.21.22\large \frac{1}{7.8.9}+\frac{1}{8.9.10}+\frac{1}{9.10.11}..............+\frac{1}{20.21.22} can be expressed as αβ\large \frac{\alpha}{\beta}, where α\alpha and β\beta are coprime, positive integers. Determine α+β\alpha+\beta.

Details and Assumptions:- a.ba.b is the same as a×ba\times b. The LCM of (7,8,9,10,.....,20,21,22)(7,8,9,10,.....,20,21,22) is 232792560232792560.

Extra Credit:- Find out what that image has to do with the answer.

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