b: \(B=\dfrac{1}{10\cdot9}+\dfrac{1}{18\cdot13}+...+\dfrac{1}{802\cdot405}\)
\(=\dfrac{2}{10\cdot18}+\dfrac{2}{18\cdot26}+...+\dfrac{2}{802\cdot810}\)
\(=\dfrac{1}{4}\left(\dfrac{8}{10\cdot18}+\dfrac{8}{18\cdot26}+...+\dfrac{8}{802\cdot810}\right)\)
\(=\dfrac{1}{4}\left(\dfrac{1}{10}-\dfrac{1}{18}+\dfrac{1}{18}-\dfrac{1}{26}+...+\dfrac{1}{802}-\dfrac{1}{810}\right)\)
\(=\dfrac{1}{4}\left(\dfrac{1}{10}-\dfrac{1}{810}\right)=\dfrac{1}{4}\cdot\dfrac{80}{810}=\dfrac{20}{810}=\dfrac{2}{81}\)
a: \(A=\dfrac{1}{2\cdot9}+\dfrac{1}{9\cdot7}+\dfrac{1}{7\cdot19}+...+\dfrac{1}{252\cdot509}\)
\(=\dfrac{2}{4\cdot9}+\dfrac{2}{9\cdot14}+\dfrac{2}{14\cdot19}+...+\dfrac{2}{504\cdot509}\)
\(=\dfrac{2}{5}\left(\dfrac{5}{4\cdot9}+\dfrac{5}{9\cdot14}+...+\dfrac{5}{504\cdot509}\right)\)
\(=\dfrac{2}{5}\left(\dfrac{1}{4}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{14}+...+\dfrac{1}{504}-\dfrac{1}{509}\right)\)
\(=\dfrac{2}{5}\left(\dfrac{1}{4}-\dfrac{1}{509}\right)=\dfrac{2}{5}\cdot\dfrac{505}{4\cdot509}=\dfrac{101}{2\cdot509}=\dfrac{101}{1018}\)
