\(\frac{3}{12}+\frac{3}{20}+\frac{3}{30}+\frac{3}{42}+\frac{3}{56}+\frac{3}{72}+\frac{3}{90}+\frac{3}{110}\)
\(=\frac{3}{3\cdot4}+\frac{3}{4\cdot5}+\frac{3}{5\cdot6}+\frac{3}{6\cdot7}+\frac{3}{7\cdot8}+\frac{3}{8\cdot7}+\frac{3}{8\cdot9}+\frac{3}{9\cdot10}+\frac{3}{10\cdot11}\)
\(=3\cdot\left(\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{6\cdot7}+\frac{1}{7\cdot8}+\frac{1}{8\cdot9}+\frac{1}{9\cdot10}+\frac{1}{10\cdot11}\right)\)
\(=3\cdot\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{10}-\frac{1}{11}\right)\)
\(=3\cdot\left(\frac{1}{3}-\frac{1}{11}\right)\)
\(=3\cdot\left(\frac{11}{33}-\frac{3}{33}\right)\)
\(=3\cdot\frac{8}{33}\)
\(=\frac{24}{33}\)
\(=\frac{8}{11}\)
