a: \(1-\frac{1}{1\cdot2}-\frac{1}{2\cdot3}-\cdots-\frac{1}{2007\cdot2008}\)
\(=1-\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+_{\ldots}+\frac{1}{2007\cdot2008}\right)\)
\(=1-\left(1-\frac12+\frac12-\frac13+\cdots+\frac{1}{2007}-\frac{1}{2008}\right)\)
\(=1-\left(1-\frac{1}{2008}\right)=\frac{1}{2008}\)
b: \(\frac89\cdot\frac{15}{16}\cdot\frac{24}{25}\cdot\ldots\cdot\frac{2499}{2500}\)
\(=\left(1-\frac19\right)\left(1-\frac{1}{16}\right)\cdot\ldots\cdot\left(1-\frac{1}{50^2}\right)\)
\(=\left(1-\frac13\right)\left(1-\frac14\right)\cdot\ldots\cdot\left(1-\frac{1}{50}\right)\left(1+\frac13\right)\left(1+\frac14\right)\cdot\ldots\cdot\left(1+\frac{1}{50}\right)\)
\(=\frac23\cdot\frac34\cdot\ldots\cdot\frac{49}{50}\cdot\frac43\cdot\frac54\cdot\ldots\cdot\frac{51}{50}=\frac{2}{50}\cdot\frac{51}{3}=\frac{17}{25}\)
c: \(\frac{4}{8\cdot13}+\frac{4}{13\cdot18}+\cdots+\frac{4}{253\cdot258}\)
\(=\frac45\left(\frac{5}{8\cdot13}+\frac{5}{13\cdot18}+\cdots+\frac{5}{253\cdot258}\right)\)
\(=\frac45\left(\frac18-\frac{1}{13}+\frac{1}{13}-\frac{1}{18}+\cdots+\frac{1}{253}-\frac{1}{258}\right)\)
\(=\frac45\left(\frac18-\frac{1}{258}\right)=\frac45\cdot\frac{250}{258\cdot8}=\frac{50}{258\cdot2}=\frac{25}{258}\)
d: \(\frac{2^2}{1\cdot3}\cdot\frac{3^2}{2\cdot4}\cdot\ldots\cdot\frac{2020^2}{2019\cdot2021}\)
\(=\frac{2\cdot3\cdot\ldots\cdot2020}{1\cdot2\cdot\ldots\cdot2019}\cdot\frac{2\cdot3\cdot\ldots\cdot2020}{3\cdot4\cdot\ldots\cdot2021}=\frac{2020}{1}\cdot\frac{2}{2021}=\frac{4040}{2021}\)
e: \(\left(\frac12+1\right)\left(\frac13+1\right)\cdot\ldots\cdot\left(\frac{1}{99}+1\right)\)
\(=\frac32\cdot\frac43\cdot\ldots\cdot\frac{100}{99}=\frac{100}{2}=50\)
g: \(\left(\frac12-1\right)\left(\frac13-1\right)\cdot\ldots\cdot\left(\frac{1}{100}-1\right)\)
\(=\frac{-1}{2}\cdot\frac{-2}{3}\cdot\ldots\cdot\frac{-99}{100}=-\frac{1}{100}\)
