\(x-\dfrac{1}{1.3}-\dfrac{1}{3.5}-...-\dfrac{1}{99.101}=\dfrac{1}{101}\)
\(\Rightarrow x-\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+...+\dfrac{1}{99.101}\right)=\dfrac{1}{101}\)
\(\Rightarrow x-\left[\dfrac{1}{2}\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+...+\dfrac{2}{99.101}\right)\right]=\dfrac{1}{101}\)
\(\Rightarrow x-\left[\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\right]=\dfrac{1}{101}\)
\(\Rightarrow x-\left[\dfrac{1}{2}\left(1-\dfrac{1}{101}\right)\right]=\dfrac{1}{101}\)
\(\Rightarrow x-\left(\dfrac{1}{2}.\dfrac{100}{101}\right)=\dfrac{1}{101}\)
\(\Rightarrow x-\dfrac{50}{101}=\dfrac{1}{101}\)
\(\Rightarrow x=\dfrac{51}{101}\)
