\(a.A=\dfrac{2}{1\cdot5}+\dfrac{2}{5\cdot9}+\dfrac{2}{9\cdot13}+..+\dfrac{2}{37\cdot41}\\ =\dfrac{1}{2}\cdot\left(\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{37\cdot41}\right)\\ =\dfrac{1}{2}\cdot\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{37}-\dfrac{1}{41}\right)\\ =\dfrac{1}{2}\cdot\left(1-\dfrac{1}{41}\right)\\ =\dfrac{1}{2}\cdot\dfrac{40}{41}\\ =\dfrac{20}{41}\\ b.B=\dfrac{1}{15}+\dfrac{1}{35}+\dfrac{1}{63}+...+\dfrac{1}{323}\\ =\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+\dfrac{1}{7\cdot9}+...+\dfrac{1}{17\cdot19}\\ =\dfrac{1}{2}\cdot\left(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}+...+\dfrac{2}{17\cdot19}\right)\\ =\dfrac{1}{2}\cdot\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{17}-\dfrac{1}{19}\right)\\ =\dfrac{1}{2}\cdot\left(\dfrac{1}{3}-\dfrac{1}{19}\right)\\ =\dfrac{1}{2}\cdot\dfrac{16}{57}\\ =\dfrac{8}{57}\)
\(C=\left(\dfrac{112}{13\cdot20}+\dfrac{112}{20\cdot27}+...+\dfrac{112}{62\cdot69}\right):\left(\dfrac{5}{9\cdot13}+\dfrac{7}{9\cdot25}+\dfrac{13}{19\cdot25}+\dfrac{31}{19\cdot69}\right)\\ =\left[16\cdot\left(\dfrac{7}{13\cdot20}+\dfrac{7}{20\cdot27}+...+\dfrac{7}{62\cdot69}\right)\right]:\left[2\cdot\left(\dfrac{5}{13\cdot18}+\dfrac{7}{18\cdot25}+\dfrac{13}{25\cdot38}+\dfrac{31}{38\cdot69}\right)\right]\\ =\left[16\cdot\left(\dfrac{1}{13}-\dfrac{1}{20}+\dfrac{1}{20}-\dfrac{1}{25}+...+\dfrac{1}{62}-\dfrac{1}{69}\right)\right]:\left[2\cdot\left(\dfrac{1}{13}-\dfrac{1}{18}+\dfrac{1}{18}-\dfrac{1}{25}+\dfrac{1}{25}-\dfrac{1}{38}+\dfrac{1}{38}-\dfrac{1}{69}\right)\right]\\ =\left[16\cdot\left(\dfrac{1}{13}-\dfrac{1}{69}\right)\right]:\left[2\cdot\left(\dfrac{1}{13}-\dfrac{1}{69}\right)\right]\\ =\dfrac{16\cdot\left(\dfrac{1}{13}-\dfrac{1}{69}\right)}{2\cdot\left(\dfrac{1}{13}-\dfrac{1}{69}\right)}\\ =\dfrac{16}{2}\\ =8\)