Đặt
\(S=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{97.99}\)
\(S=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)\)
\(S=\dfrac{1}{2}\left(1-\dfrac{1}{99}\right)\)
\(S=\dfrac{49}{99}\)
\(\Rightarrow\dfrac{1}{x}-\dfrac{1}{9999}=\dfrac{49}{99}\)
\(\Rightarrow\dfrac{1}{x}=\dfrac{50}{101}\Leftrightarrow50x=101\Leftrightarrow x=\dfrac{101}{50}\)
Đặt \(M=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{97.99}\)
\(M=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)\)
\(M=\dfrac{1}{2}\left(1-\dfrac{1}{99}\right)\)
\(M=\dfrac{1}{2}\cdot\dfrac{98}{99}\)
\(M=\dfrac{49}{99}\)