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$\large \frac{1}{7.8.9}+\frac{1}{8.9.10}+\frac{1}{9.10.11}..............+\frac{1}{20.21.22}$ 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.