\(\frac{1}{9.10}-\frac{1}{8.9}-...-\frac{1}{1.2}\)
\(=-\left(\frac{1}{9.10}+\frac{1}{8.9}+...+\frac{1}{1.2}\right)\)
\(=-\left(\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{9.10}\right)\)
\(=-\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{9}-\frac{1}{10}\right)\)
\(=-\left(1-\frac{1}{10}\right)\)
\(=-\frac{9}{10}\)
\(\frac{1}{9.10}-\frac{1}{8.9}-....-\frac{1}{1.2}\)
\(=\frac{1}{9.10}-\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{8.9}\right)\)
\(\Rightarrow\frac{1}{9.10}-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{8}-\frac{1}{9}\right)\)
\(=\frac{1}{90}-\left(1-\frac{1}{9}\right)=\frac{1}{90}-\frac{80}{90}=\frac{-79}{90}\)
