\(\dfrac{5}{1\cdot2\cdot3}+\dfrac{5}{2\cdot3\cdot4}+...+\dfrac{5}{99\cdot100\cdot101}\)
\(=5\cdot\left(\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+...+\dfrac{1}{99\cdot100\cdot101}\right)\)
\(=5\cdot\dfrac{1}{2}\cdot\left(\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{99\cdot100}-\dfrac{1}{100\cdot101}\right)\)
\(=\dfrac{5}{2}\cdot\left(\dfrac{1}{1\cdot2}-\dfrac{1}{100\cdot101}\right)\)
\(=\dfrac{5}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{10100}\right)\)
\(=\dfrac{5}{2}\cdot\dfrac{5049}{10100}\)
\(=\dfrac{25245}{20200}=\dfrac{5049}{4040}\)
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