Đặt \(A=\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}\)
\(\Rightarrow2A=\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\)
\(\Rightarrow2A=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\)
\(\Rightarrow2A=\dfrac{1}{3}-\dfrac{1}{101}\)
\(\Rightarrow2A=\dfrac{98}{303}\)
\(\Rightarrow A=\dfrac{98}{303}.\dfrac{1}{2}\)
\(\Rightarrow A=\dfrac{49}{303}\)
\(\dfrac{1}{3.5}+\dfrac{1}{5.7}+.....+\dfrac{1}{99.101}\)
\(=\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+.....+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{100}\right)\)
\(=\dfrac{1}{2}.\dfrac{97}{300}=\dfrac{97}{600}\)
