Question

1/1.2.3+1/2.3.4+...+1/8.9.10

Answer 1

\(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}...+\frac{2}{8.9.10}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{90}\right)=\frac{11}{45}\)

Answer 2

     \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{8.9.10}\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{90}\right)\)

\(=\frac{1}{2}.\frac{22}{45}\)

\(=\frac{11}{45}\)