a: \(A=\dfrac{3^2}{2\cdot5}+\dfrac{3^2}{5\cdot8}+...+\dfrac{3^2}{14\cdot17}\)
\(=3\cdot\left(\dfrac{3}{2\cdot5}+\dfrac{3}{5\cdot8}+...+\dfrac{3}{14\cdot17}\right)\)
\(=3\left(\dfrac{1}{2}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{8}+...+\dfrac{1}{14}-\dfrac{1}{17}\right)\)
\(=3\left(\dfrac{1}{2}-\dfrac{1}{17}\right)=3\cdot\dfrac{15}{34}=\dfrac{45}{34}\)
b: \(B=\dfrac{4}{11\cdot16}+\dfrac{4}{16\cdot21}+...+\dfrac{4}{61\cdot66}\)
\(=\dfrac{4}{5}\left(\dfrac{5}{11\cdot16}+\dfrac{5}{16\cdot21}+...+\dfrac{5}{61\cdot66}\right)\)
\(=\dfrac{4}{5}\left(\dfrac{1}{11}-\dfrac{1}{16}+\dfrac{1}{16}-\dfrac{1}{21}+...+\dfrac{1}{61}-\dfrac{1}{66}\right)\)
\(=\dfrac{4}{5}\left(\dfrac{1}{11}-\dfrac{1}{66}\right)=\dfrac{4}{5}\cdot\dfrac{5}{66}=\dfrac{4}{66}=\dfrac{2}{33}\)
c: \(C=\dfrac{5^2}{1\cdot6}+\dfrac{5^2}{6\cdot11}+...+\dfrac{5^2}{26\cdot31}\)
\(=5\left(\dfrac{5}{1\cdot6}+\dfrac{5}{6\cdot11}+...+\dfrac{5}{26\cdot31}\right)\)
\(=5\left(1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+...+\dfrac{1}{26}-\dfrac{1}{31}\right)=5\left(1-\dfrac{1}{31}\right)=5\cdot\dfrac{30}{31}=\dfrac{150}{31}\)