\(\frac{a}{b}=\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{99.100}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{100}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{100}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)=\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)
\(=\left(\frac{1}{51}+\frac{1}{100}\right)+...+\left(\frac{1}{75}+\frac{1}{76}\right)=\frac{151}{100.51}+...+\frac{151}{75.76}\)
\(=151.\left(\frac{1}{51.100}+...+\frac{1}{75.76}\right)\)
gọi \(\frac{1}{51.100}+...+\frac{1}{75.76}=\frac{c}{d}\Rightarrow\frac{a}{b}=\frac{c}{d}.151=\frac{151c}{d}\)
=>a chia hết cho 151
=>đpcm
