\(\frac{1}{7\cdot12}+\frac{1}{12\cdot17}+...+\frac{1}{2017\cdot2022}\)
\(=\frac{1}{5}\left(\frac{5}{7\cdot12}+\frac{5}{12\cdot17}+...+\frac{5}{2017\cdot2022}\right)\)
\(=\frac{1}{5}\left(\frac{1}{7}-\frac{1}{2}+\frac{1}{12}-\frac{1}{17}+...+\frac{1}{2017}-\frac{1}{2022}\right)\)
\(=\frac{1}{5}\left(\frac{1}{7}-\frac{1}{2022}\right)\)
\(=\frac{1}{4}\cdot\frac{2015}{14154}\)
\(=\frac{2015}{56616}\)
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