Question

\(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...\frac{1}{20}\)= ??

Answer 1

\(=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{6}\right)+\left(\frac{1}{6}-\frac{1}{8}\right)+\left(\frac{1}{8}-\frac{1}{10}\right)+\left(\frac{1}{10}-\frac{1}{12}\right)+\left(\frac{1}{12}-\frac{1}{14}\right)+\left(\frac{1}{14}-\frac{1}{16}\right)+\left(\frac{1}{16}-\frac{1}{18}\right)+\left(\frac{1}{18}-\frac{1}{20}\right)\)\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+\frac{1}{8}-\frac{1}{10}+\frac{1}{10}-\frac{1}{12}+\frac{1}{12}-\frac{1}{14}+\frac{1}{14}-\frac{1}{16}+\frac{1}{16}-\frac{1}{18}+\frac{1}{18}-\frac{1}{20}\)\(1-\frac{1}{20}=\frac{19}{20}\)