\(a.A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{99\cdot100}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\\ =1-\dfrac{1}{100}\\ =\dfrac{99}{100}\\ b.B=\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)...\left(1-\dfrac{1}{n+1}\right)\\ =\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot\dfrac{3}{4}\cdot...\cdot\dfrac{n}{n+1}\\ =\dfrac{2\cdot3\cdot4\cdot...\cdot n}{\left(2\cdot3\cdot4\cdot...\cdot n\right)\cdot\left(n+1\right)}\\ =\dfrac{1}{n+1}\)
\(c.C=-66\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{11}\right)+124\cdot\left(-37\right)+63\cdot\left(-124\right)\\ =-66\cdot\left(\dfrac{1}{6}+\dfrac{1}{11}\right)+124\cdot\left(-37-63\right)\\ =-66\cdot\dfrac{17}{66}+124\cdot-100\\ =-17-12400\\ =-12417\\ d.D=\dfrac{7}{4}\cdot\left(\dfrac{33}{12}+\dfrac{3333}{2020}+\dfrac{333333}{303030}+\dfrac{33333333}{42424242}\right)\\ =\dfrac{7}{4}\left(\dfrac{33}{12}+\dfrac{33\cdot101}{20\cdot101}+\dfrac{33\cdot10101}{30\cdot10101}+\dfrac{33\cdot1010101}{42\cdot1010101}\right)\\ =\dfrac{7}{4}\cdot\left(\dfrac{33}{12}+\dfrac{33}{20}+\dfrac{33}{30}+\dfrac{33}{42}\right)\\ =\dfrac{7}{4}\cdot33\cdot\left(\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+\dfrac{1}{6\cdot7}\right)\\ =\dfrac{7}{4}\cdot33\cdot\left(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\right)\\ =\dfrac{7}{4}\cdot33\cdot\left(\dfrac{1}{3}-\dfrac{1}{7}\right)\\ =\dfrac{7}{4}\cdot33\cdot\dfrac{4}{21}\\ =11\)
