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Algebra Level 4

\(\large \frac{1}{7.8.9}+\frac{1}{8.9.10}+\frac{1}{9.10.11}..............+\frac{1}{20.21.22}\) can be expressed as \(\large \frac{\alpha}{\beta}\), where \(\alpha\) and \(\beta\) are coprime, positive integers. Determine \(\alpha+\beta\).

Details and Assumptions:- \(a.b\) is the same as \(a\times b\). The LCM of \((7,8,9,10,.....,20,21,22)\) is \(232792560\).

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

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