\(=\frac{49\times50-1}{4\times98\times100}\)

Đặt biểu thức là A

\(4xA=\frac{4}{2x4x6}+\frac{4}{4x6x8}+\frac{4}{6x8x10}+\frac{4}{8x10x12}+...+\frac{4}{94x96x98}+\frac{4}{96x98x100}\)

\(4xA=\frac{6-2}{2x4x6}+\frac{8-4}{4x6x8}+\frac{10-6}{6x8x10}+\frac{12-8}{8x10x12}+...+\frac{98-94}{94x96x98}+\frac{100-96}{96x98x100}\)

\(4xA=\frac{1}{2x4}-\frac{1}{4x6}+\frac{1}{4x6}-\frac{1}{6x8}+\frac{1}{6x8}-\frac{1}{8x10}+...+\frac{1}{94x96}-\frac{1}{96x98}+\frac{1}{96x98}-\frac{1}{98x100}\)

\(4xA=\frac{1}{2x4}-\frac{1}{98x100}=\frac{49x50-1}{98x100}\Rightarrow A=\frac{49x50-1}{4x98x100}\)

\(\frac{1}{2.4.6}+\frac{1}{4.6.8}+\frac{1}{6.8.10}+..+\frac{1}{50.52.54}\)

\(=\frac{1}{4}.\left(\frac{1}{2.4}-\frac{1}{4.6}+\frac{1}{4.6}-\frac{1}{6.8}+....+\frac{1}{50.52}-\frac{1}{52.54}\right)\)

\(=\frac{1}{4}.\left(\frac{1}{2.4}-\frac{1}{52.54}\right)\)

\(=\frac{1}{4}.\frac{175}{1404}=\frac{175}{5616}\)

\(A=\frac{5}{4}.\left(\frac{4}{2.4.6}+\frac{4}{4.6.8}+\frac{4}{6.8.10}+....+\frac{4}{16.18.20}\right)\)

     \(=\frac{5}{4}.\left(\frac{1}{2.4}-\frac{1}{4.6}+\frac{1}{4.6}-\frac{1}{6.8}+\frac{1}{6.8}-\frac{1}{8.10}+....+\frac{1}{16.18}-\frac{1}{18.20}\right)\)

    \(=\frac{5}{4}.\left(\frac{1}{2.4}-\frac{1}{18.20}\right)=\frac{5}{4}.\frac{11}{90}=\frac{11}{72}\)

\(A=\frac{1}{3}.\left(\frac{3}{2.4.6}+\frac{3}{4.6.8}+...+\frac{3}{22.24.26}\right)\)

\(A=\frac{1}{3}.\left(\frac{1}{2.4}-\frac{1}{4.6}+\frac{1}{4.6}-\frac{1}{6.8}+...+\frac{1}{22.24}-\frac{1}{24.26}\right)\)

\(A=\frac{1}{3}.\left(\frac{1}{2.4}-\frac{1}{24.26}\right)\)

\(A=\frac{1}{3}.\frac{77}{624}\)

\(A=\frac{77}{1872}\)