Ta có:
\(M=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}\)
\(M=\frac{1}{\left(2\times3\right):2}+\frac{1}{\left(3\times4\right):2}+\frac{1}{\left(4\times5\right):2}+\frac{1}{\left(5\times6\right):2}\)
\(M=\frac{2}{2\times3}+\frac{2}{3\times4}+\frac{2}{4\times5}+\frac{2}{5\times6}\)
\(M:2=\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+\frac{1}{5\times6}\)
\(M:2=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\)
\(M:2=\frac{1}{2}-\frac{1}{6}\)
\(M:2=\frac{1}{3}\)
\(\Rightarrow M=\frac{1}{3}\times2\)
\(\Rightarrow M=\frac{2}{3}\)