a: \(\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\cdots+\frac{1}{26\cdot27}\)
\(=\frac13-\frac14+\frac14-\frac15+\cdots+\frac{1}{26}-\frac{1}{27}\)
\(=\frac13-\frac{1}{27}=\frac{8}{27}\)
b: \(\left(\frac12-1\right)\left(\frac13-1\right)\cdot\ldots\cdot\left(\frac{1}{2019}-1\right)\)
\(=\frac{-1}{2}\cdot\frac{-2}{3}\cdot\ldots\cdot\frac{-2018}{2019}=1\cdot\frac22\cdot\frac33\cdot\ldots\cdot\frac{2018}{2018}\cdot\frac{1}{2019}=\frac{1}{2019}\)
c: \(\left(1-\frac14\right)\left(1-\frac19\right)\cdot\ldots\cdot\left(1-\frac{1}{81}\right)\cdot\left(1-\frac{1}{100}\right)\)
\(=\left(1-\frac12\right)\left(1-\frac13\right)\cdot\ldots\cdot\left(1-\frac{1}{10}\right)\left(1+\frac12\right)\left(1+\frac13\right)\cdot\ldots\cdot\left(1+\frac{1}{10}\right)\)
\(=\frac12\cdot\frac23\cdot\ldots\cdot\frac{9}{10}\cdot\frac32\cdot\frac43\cdot\ldots\cdot\frac{11}{10}=\frac{1}{10}\cdot\frac{11}{2}=\frac{11}{20}\)
d: \(\left(-1\frac12\right)\cdot\left(-1\frac13\right)\cdot\ldots\cdot\left(-1\frac{1}{2020}\right)\)
\(=\frac{-3}{2}\cdot\frac{-4}{3}\cdot\ldots\cdot\frac{-2021}{2020}\)
\(=-\frac32\cdot\frac43\cdot\ldots\cdot\frac{2021}{2020}=-\frac{2021}{2}\)
