\(A=\frac{10^2}{1\cdot6}+\frac{10^2}{6\cdot11}+...+\frac{10^2}{61\cdot66}=\left(\frac{5}{1\cdot6}+\frac{5}{6\cdot11}+...+\frac{5}{61\cdot66}\right)\cdot20\)
\(=\left(1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+...+\frac{1}{61}-\frac{1}{66}\right)\cdot20\)
\(=\left[\left(1-\frac{1}{66}\right)+\left(\frac{1}{6}-\frac{1}{6}\right)+...+\left(\frac{1}{61}-\frac{1}{61}\right)\right]\cdot20\)
\(=\left[\left(\frac{66}{66}-\frac{1}{66}\right)+0+...+0\right]\cdot20=\frac{65}{66}\cdot20=\frac{65\cdot20}{66}=\frac{65\cdot10}{33}=\frac{650}{33}\)
\(A=\frac{10^2}{1.6}+\frac{10^2}{6.11}+...+\frac{10^2}{61.66}\)
\(=10^2.\left(\frac{1}{1.6}+\frac{1}{6.11}+...+\frac{1}{61.66}\right)\)
\(=10^2.5.\left(\frac{1}{1}-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+...+\frac{1}{61}-\frac{1}{66}\right)\)
\(=500.\left(1-\frac{1}{66}\right)\)
\(=500.\frac{65}{66}\)
\(=\frac{16250}{33}\)