\(A=\dfrac{7}{1\cdot2\cdot3\cdot4}+\dfrac{7}{2\cdot3\cdot4\cdot5}+...+\dfrac{7}{98\cdot99\cdot100\cdot101}\\ =\dfrac{7}{3}\cdot\left(\dfrac{3}{1\cdot2\cdot3\cdot4}+\dfrac{3}{2\cdot3\cdot4\cdot5}+...+\dfrac{3}{98\cdot99\cdot100\cdot101}\right)\\ =\dfrac{7}{3}\cdot\left(\dfrac{4-1}{1\cdot2\cdot3\cdot4}+\dfrac{5-2}{2\cdot3\cdot4\cdot5}+...+\dfrac{101-98}{98\cdot99\cdot100\cdot101}\right)\\ =\dfrac{7}{3}\cdot\left(\dfrac{4}{1\cdot2\cdot3\cdot4}-\dfrac{1}{1\cdot2\cdot3\cdot4}+\dfrac{5}{2\cdot3\cdot4\cdot5}-\dfrac{2}{2\cdot3\cdot4\cdot5}+...+\dfrac{101}{98\cdot99\cdot100\cdot101}-\dfrac{98}{98\cdot99\cdot100\cdot101}\right)\\ =\dfrac{7}{3}\cdot\left(\dfrac{1}{1\cdot2\cdot3}-\dfrac{1}{2\cdot3\cdot4}+\dfrac{1}{2\cdot3\cdot4}-\dfrac{1}{3\cdot4\cdot5}+...+\dfrac{1}{98\cdot99\cdot100}-\dfrac{1}{99\cdot100\cdot101}\right)\\ =\dfrac{7}{3}\cdot\left(\dfrac{1}{1\cdot2\cdot3}-\dfrac{1}{99\cdot100\cdot101}\right)\\ =\dfrac{7}{3}\cdot\left(\dfrac{1}{6}-\dfrac{1}{999900}\right)\\ =\dfrac{7}{3}\cdot\dfrac{166649}{999900}=\dfrac{1166543}{2999700}\)
