Ta có: \(M=\frac{4}{1.2.3}+\frac{4}{2.3.4}+...+\frac{4}{8.9.10}\)
\(M=4\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{8.9.10}\right)\)
\(M=4.\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\right)\)
\(M=2\left(\frac{1}{2}-\frac{1}{90}\right)=1-\frac{1}{45}=\frac{44}{45}\)
Vậy \(M=\frac{44}{45}\)
\(M=\frac{4}{1.2.3}+\frac{4}{2.3.4}+...+\frac{4}{8.9.10}\)
\(=2.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{8.9.10}\right)\)
\(=2.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\right)\)
\(=2.\left(\frac{1}{1.2}-\frac{1}{9.10}\right)\)
\(=2.\left(\frac{1}{2}-\frac{1}{90}\right)\)
\(=2.\frac{22}{45}\)
\(=\frac{44}{45}\)