a) \(\frac{3}{4\times9}+\frac{3}{9\times14}+...+\frac{3}{54\times59}+\frac{3}{59\times64}\)
\(=\frac{3}{5}\times\left(\frac{5}{4\times9}+\frac{5}{9\times14}+...+\frac{5}{59\times64}\right)\)
\(=\frac{3}{5}\times\left(\frac{9-4}{4\times9}+\frac{14-9}{9\times14}+...+\frac{64-59}{59\times64}\right)\)
\(=\frac{3}{5}\times\left(\frac{1}{4}-\frac{1}{9}+\frac{1}{9}-\frac{1}{14}+...+\frac{1}{59}-\frac{1}{64}\right)\)
\(=\frac{3}{5}\times\left(\frac{1}{4}-\frac{1}{64}\right)\)
\(=\frac{9}{64}\)
b) \(\frac{2}{8\times11}+\frac{2}{11\times14}+...+\frac{2}{23\times26}+\frac{2}{26\times29}\)
\(=\frac{2}{3}\times\left(\frac{3}{8\times11}+\frac{3}{11\times14}+...+\frac{3}{26\times29}\right)\)
\(=\frac{2}{3}\times\left(\frac{11-8}{8\times11}+\frac{14-11}{11\times14}+...+\frac{29-26}{26\times29}\right)\)
\(=\frac{2}{3}\times\left(\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+...+\frac{1}{26}-\frac{1}{29}\right)\)
\(=\frac{2}{3}\times\left(\frac{1}{8}-\frac{1}{29}\right)\)
\(=\frac{7}{116}\)
