Đặt \(A=\dfrac{12}{2\times4}+\dfrac{12}{4\times6}+...+\dfrac{12}{100\times102}\)
\(\Rightarrow\dfrac{1}{6}A=\dfrac{2}{2\times4}+\dfrac{2}{4\times6}+...+\dfrac{2}{100\times102}\)
\(\Rightarrow\dfrac{1}{6}A=\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{100}-\dfrac{1}{102}\)
\(\Rightarrow\dfrac{1}{6}A=\dfrac{1}{2}-\dfrac{1}{102}\)
\(\Rightarrow\dfrac{1}{6}A=\dfrac{25}{51}=\dfrac{150}{51}\)
\(=12\cdot\left(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}+...+\dfrac{1}{100\cdot102}\right)\\ =12\cdot\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{100}-\dfrac{1}{102}\right)\\ =12\cdot\left(\dfrac{1}{2}-\dfrac{1}{102}\right)=12\cdot\dfrac{50}{102}=\dfrac{100}{17}\)
