\(\frac{1}{100.99}-\frac{1}{99.98}-...-\frac{1}{2.1}\)
\(\frac{1}{100-99}-\left(\frac{1}{99.98}+\frac{1}{98.97}+..+\frac{1}{2.1}\right)\)
\(\frac{1}{100-99}-\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{98.99}\right)\)
\(\frac{1}{100.99}-\left(\frac{1}{1}-\frac{1}{2}+...+\frac{1}{98}-\frac{1}{99}\right)\)
\(\frac{1}{100.99}-\left(\frac{1}{1}-\frac{1}{99}\right)\)
\(\frac{1}{99}-\frac{1}{100}-\frac{98}{99}\)
\(-\frac{97}{99}-\frac{1}{100}\)
\(-\frac{9799}{9900}\)
\(\frac{1}{100\cdot99}-\frac{1}{99\cdot98}-...-\frac{1}{2\cdot1}\)
\(=\frac{1}{100\cdot99}-\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{98\cdot99}\right)\)
\(=\frac{1}{99\cdot100}-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{98}-\frac{1}{99}\right)\)
\(=\frac{1}{9900}-\frac{98}{99}\)
\(=\frac{-9799}{9900}\)
