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\(\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\frac{5}{6}-\frac{6}{7}+\frac{4}{5}-\frac{3}{4}+\frac{2}{3}-\frac{1}{2}\)
= \(\left(\frac{1}{2}-\frac{1}{2}\right)+\left(\frac{-2}{3}+\frac{2}{3}\right)+\left(\frac{3}{4}-\frac{3}{4}\right)+\left(\frac{-4}{5}+\frac{4}{5}\right)+\frac{5}{6}-\frac{6}{7}\)
= \(\frac{5}{6}-\frac{6}{7}\)
= \(\frac{-1}{42}\)
\(\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\frac{5}{6}-\frac{6}{5}+\frac{4}{5}-\frac{3}{4}+\frac{2}{3}-\frac{1}{2}\)
\(=\left(\frac{1}{2}-\frac{1}{2}\right)+\left(\frac{-2}{3}+\frac{2}{3}\right)+\left(\frac{3}{4}-\frac{3}{4}\right)+\left(\frac{-4}{5}+\frac{4}{5}\right)+\frac{5}{6}-\frac{6}{7}\)
\(=\frac{5}{6}-\frac{6}{7}\)
\(=\frac{-1}{42}\)
\(\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}-\frac{6}{7}-\frac{5}{6}+\frac{4}{5}-\frac{3}{4}+\frac{2}{3}-\frac{1}{2}\)
\(=\left(\frac{1}{2}-\frac{1}{2}\right)-\left(\frac{2}{3}-\frac{2}{3}\right)+\left(\frac{3}{4}-\frac{3}{4}\right)-\left(\frac{4}{5}-\frac{4}{5}\right)-\left(\frac{6}{7}+\frac{5}{6}\right)\)
\(=-\frac{71}{42}\)