Question

1/2×3×4+1/3×4×5+1/4×5×6+.......+1/10×11×12=??

Answer 1

\(S=\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+\frac{1}{4\cdot5\cdot6}+...+\frac{1}{10\cdot11\cdot12}\)

\(2S=\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+\frac{2}{4\cdot5\cdot6}+...+\frac{2}{10\cdot11\cdot12}\)

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

\(2S=\frac{1}{2\cdot3}-\frac{1}{11\cdot12}\)

\(2S=\frac{1}{6}-\frac{1}{132}\)

\(2S=\frac{21}{132}\)

\(S=\frac{21}{132}\div2\)

\(S=\frac{21}{264}\)