\(A=5\left(\dfrac{1}{8\times9}+\dfrac{1}{9\times10}+...+\dfrac{1}{99\times100}\right)\\ A=5\left(\dfrac{1}{8}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{10}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\\ A=5\left(\dfrac{1}{8}-\dfrac{1}{100}\right)=5\times\dfrac{23}{200}=\dfrac{23}{40}\\ B=\dfrac{7}{2}\left(\dfrac{2}{5\times7}+\dfrac{2}{7\times9}+...+\dfrac{2}{87\times89}\right)\\ B=\dfrac{7}{2}\left(\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{87}-\dfrac{1}{89}\right)\\ B=\dfrac{7}{2}\left(\dfrac{1}{5}-\dfrac{1}{89}\right)=\dfrac{7}{2}\times\dfrac{84}{445}=\dfrac{294}{445}\)
\(C=2\left(\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{72}\right)\\ C=2\left(\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{8\times9}\right)\\ C=2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{8}-\dfrac{1}{9}\right)\\ C=2\left(\dfrac{1}{2}-\dfrac{1}{9}\right)=2\times\dfrac{7}{18}=\dfrac{7}{9}\\ D=\dfrac{3}{2}\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{90}\right)\\ D=\dfrac{3}{2}\left(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+...+\dfrac{1}{9\times10}\right)\\ D=\dfrac{3}{2}\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{9}-\dfrac{1}{10}\right)\\ D=\dfrac{3}{2}\left(1-\dfrac{1}{10}\right)=\dfrac{3}{2}\times\dfrac{9}{10}=\dfrac{27}{20}\)
1) \(A=\dfrac{5}{8\times9}+\dfrac{5}{9\times10}+...+\dfrac{5}{99\times100}\)
\(\Rightarrow A=5\left(\dfrac{1}{8\times9}+\dfrac{1}{9\times10}+...+\dfrac{1}{99\times100}\right)\)
\(\Rightarrow A=5\left(\dfrac{1}{8}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{10}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(\Rightarrow A=5\left(\dfrac{1}{8}-\dfrac{1}{100}\right)\)
\(\Rightarrow A=\dfrac{5.23}{200}\)
\(\Rightarrow A=\dfrac{23}{40}\)
b) \(B=\dfrac{7}{5\times7}+\dfrac{7}{7\times9}+...+\dfrac{7}{89\times89}\)
\(\Rightarrow B=7\left(\dfrac{1}{5\times7}+\dfrac{1}{7\times9}+...+\dfrac{1}{87\times89}\right)\)
\(\Rightarrow2B=7\left(\dfrac{2}{5\times7}+\dfrac{2}{7\times9}+...+\dfrac{2}{87\times89}\right)\)
\(\Rightarrow2B=7\left(\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{87}-\dfrac{1}{89}\right)\)
\(\Rightarrow2B=7\left(\dfrac{1}{5}-\dfrac{1}{89}\right)\)
\(\Rightarrow2B=\dfrac{7.84}{445}\)
\(\Rightarrow2B=\dfrac{588}{445}\)
\(\Rightarrow B=\dfrac{294}{445}\)