1/100.99 - 1/99.98 - 1/98.97 - ... - 1/3.2 - 1/2.1
= 1/100 - (1/100.99 - 1/99.98 - 1/98.97 - ... - 1/3.2 - 1/2.1)
= 1/100 - (1/1.2 + 1/2.3 + ... + 1/97.98 + 1/98.99 + 1/99.100)
= 1/100 - (1 - 1/2 + 1/2 - 1/3 + ... + 1/97 - 1/98 + 1/98 - 1/99 + 1/99 - 1/100)
= 1/100 - (1 - 1/100)
= 1/100 - 99/100
= -49/50
\(\frac{1}{100.99}-\frac{1}{99.98}-\frac{1}{98.97}-...-\frac{1}{3.2}-\frac{1}{2.1}\)
\(=-\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{98.99}+\frac{1}{99.100}\right)\)
\(=-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\right)\)
\(=-\left(1-\frac{1}{100}\right)\)
\(=-\frac{99}{100}\)
\(\frac{1}{100.99}-\frac{1}{99.98}-\frac{1}{98.97}-...-\frac{1}{3.2}-\frac{1}{2.1}\)
\(=-\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{98.99}+\frac{1}{99.100}\right)\)
\(=-\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\right)\)
\(=-\left(\frac{1}{1}-\frac{1}{100}\right)\)
\(=-\frac{99}{100}\)
