\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}\)\(\frac{1}{6.7}\)=\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}\)
=\(\frac{1}{1}-\frac{1}{7}=\frac{6}{7}\)
1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42
=1/1x2 + 1/2x3 + 1/3x4 + 1/4x5 + 1/5x6 + 1/6x7
=1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7
= 1 - 1/7
= 6/7
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}\)
\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}\)
\(=1-\frac{1}{7}\)
\(=\frac{6}{7}\)
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}\)
\(=\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+\frac{1}{5\times6}+\frac{1}{6\times7}\)
\(=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)+\left(\frac{1}{6}-\frac{1}{7}\right)\)
\(=1-\left(\frac{1}{2}-\frac{1}{2}\right)-\left(\frac{1}{3}-\frac{1}{3}\right)-\left(\frac{1}{4}-\frac{1}{4}\right)-\left(\frac{1}{5}-\frac{1}{5}\right)-\left(\frac{1}{6}-\frac{1}{6}\right)-\frac{1}{7}\)
=1-1/7
=6/7
\(\frac{1}{2}\)+\(\frac{1}{6}\)+\(\frac{1}{12}\)+\(\frac{1}{20}\)+\(\frac{1}{30}\)+\(\frac{1}{42}\)
=\(\frac{1}{1x2}\)+\(\frac{1}{2x3}\)+\(\frac{1}{3x4}\)+\(\frac{1}{4x5}\)+\(\frac{1}{5x6}\)+\(\frac{1}{6x7}\)
=1-\(\frac{1}{2}\)+\(\frac{1}{2}-\frac{1}{3}\)+\(\frac{1}{3}-\frac{1}{4}\)+\(\frac{1}{4}-\frac{1}{5}\)+\(\frac{1}{5}-\frac{1}{6}\)+\(\frac{1}{6}-\frac{1}{7}\)
=1-\(\frac{1}{7}\)
=\(\frac{6}{7}\)
