a: \(A=\frac{\frac37-\frac{3}{17}+\frac{3}{37}}{\frac57-\frac{5}{17}+\frac{5}{37}}+\frac{\frac12-\frac13+\frac14-\frac15}{\frac72-\frac73+\frac74-\frac75}\)
\(=\frac{3\left(\frac17-\frac{1}{17}+\frac{1}{37}\right)}{5\left(\frac17-\frac{1}{17}+\frac{1}{37}\right)}+\frac{\frac12-\frac13+\frac14-\frac15}{7\left(\frac12-\frac13+\frac14-\frac15\right)}\)
\(=\frac35+\frac17=\frac{26}{35}\)
b: \(\frac{7}{3\cdot6}+\frac{7}{6\cdot9}+\cdots+\frac{7}{66\cdot69}\)
\(=\frac73\left(\frac{3}{6\cdot9}+\frac{3}{9\cdot12}+\cdots+\frac{3}{66\cdot69}\right)\)
\(=\frac73\left(\frac16-\frac19+\frac19-\frac{1}{12}+\cdots+\frac{1}{66}-\frac{1}{69}\right)\)
\(=\frac73\left(\frac16-\frac{1}{69}\right)=\frac73\cdot\frac{69-6}{69\cdot6}=\frac73\cdot\frac{63}{69\cdot6}=\frac{7\cdot21}{69\cdot6}=\frac{7\cdot7}{69\cdot2}=\frac{49}{138}\)
\(\frac{5}{1\cdot3}+\frac{5}{3\cdot5}+\cdots+\frac{5}{67\cdot69}\)
\(=\frac52\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\cdots+\frac{2}{67\cdot69}\right)\)
\(=\frac52\left(1-\frac13+\frac13-\frac15+\cdots+\frac{1}{67}-\frac{1}{69}\right)\)
\(=\frac52\left(1-\frac{1}{69}\right)=\frac52\cdot\frac{68}{69}=5\cdot\frac{34}{69}=\frac{170}{69}\)
Ta có: \(\left(\frac{7}{3\cdot6}+\frac{7}{6\cdot9}+\cdots+\frac{7}{66\cdot69}\right):\left(\frac{5}{1\cdot3}+\frac{5}{3\cdot5}+\cdots+\frac{5}{67\cdot69}\right)\)
\(=\frac{49}{138}:\frac{170}{69}=\frac{49}{138}\cdot\frac{69}{170}=\frac{49}{2\cdot170}=\frac{49}{340}\)
