\(S=\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+\frac{1}{4\cdot5\cdot6}+...+\frac{1}{10\cdot11\cdot12}\)
\(2S=\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+\frac{2}{4\cdot5\cdot6}+...+\frac{2}{10\cdot11\cdot12}\)
\(2S=\left(\frac{1}{2\cdot3}-\frac{1}{3\cdot4}\right)+\left(\frac{1}{3\cdot4}-\frac{1}{4\cdot5}\right)+\left(\frac{1}{4\cdot5}-\frac{1}{5\cdot6}\right)+...+\left(\frac{1}{10\cdot11}-\frac{1}{11\cdot12}\right)\)
\(2S=\frac{1}{2\cdot3}-\frac{1}{11\cdot12}\)
\(2S=\frac{1}{6}-\frac{1}{132}\)
\(2S=\frac{21}{132}\)
\(S=\frac{21}{132}\div2\)
\(S=\frac{21}{264}\)