Ta có: \(A=\frac{1}{1\times300}+\frac{1}{2\times301}+\cdots+\frac{1}{101\times400}\)
\(=\frac{1}{299}\times\left(\frac{299}{1\times300}+\frac{299}{2\times301}+\cdots+\frac{299}{101\times400}\right)\)
\(=\frac{1}{299}\times\left(1-\frac{1}{300}+\frac12-\frac{1}{301}+\cdots+\frac{1}{101}-\frac{1}{400}\right)\)
\(=\frac{1}{299}\times\left(1+\frac12+\cdots+\frac{1}{101}-\frac{1}{300}-\frac{1}{301}-\cdots-\frac{1}{400}\right)\)
Ta có: \(B=\frac{1}{1\times102}+\frac{1}{2\times103}+\cdots+\frac{1}{299\times400}\)
\(=\frac{1}{101}\times\left(\frac{101}{1\times102}+\frac{101}{2\times103}+\cdots+\frac{101}{299\times400}\right)\)
\(=\frac{1}{101}\times\left(1-\frac{1}{102}+\frac12-\frac{1}{103}+\cdots+\frac{1}{299}-\frac{1}{400}\right)\)
\(=\frac{1}{101}\times\left(1+\frac12+\cdots+\frac{1}{299}-\frac{1}{102}-\frac{1}{103}-\cdots-\frac{1}{400}\right)\)
\(=\frac{1}{101}\times\left(1+\frac12+\cdots+\frac{1}{101}-\frac{1}{300}-\frac{1}{301}-\cdots-\frac{1}{400}\right)\)
Do đó: \(\frac{A}{B}=\frac{1}{299}:\frac{1}{101}=\frac{101}{299}\)
