\(\dfrac{4}{2\cdot4}+\dfrac{4}{4\cdot6}+\dfrac{4}{6\cdot8}+...+\dfrac{4}{16\cdot18}+\dfrac{4}{18\cdot20}\)
\(=2\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+\dfrac{2}{6\cdot8}+...+\dfrac{2}{16\cdot18}+\dfrac{2}{18\cdot20}\right)\)
\(=2\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{16}-\dfrac{1}{18}+\dfrac{1}{18}-\dfrac{1}{20}\right)\)
\(=2\left(1-\dfrac{1}{20}\right)\)
\(=2\left(\dfrac{20}{20}-\dfrac{1}{20}\right)\)
\(=2\cdot\dfrac{19}{20}\)
\(=\dfrac{19}{10}\)
\(\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+.....+\dfrac{4}{16.18}+\dfrac{4}{18.20}\)
\(=\dfrac{4}{2}\left(\dfrac{1}{2.4}+\dfrac{1}{4.6}+\dfrac{1}{6.8}+.....+\dfrac{1}{16.18}+\dfrac{1}{18.20}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+.....+\dfrac{1}{16}-\dfrac{1}{18}+\dfrac{1}{18}-\dfrac{1}{20}\right)\)\(=2\left(1-\dfrac{1}{20}\right)=2.\dfrac{19}{20}=\dfrac{19}{10}\)
\(\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+..................+\dfrac{4}{16.18}+\dfrac{4}{18.20}\)
\(=2\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...................+\dfrac{2}{16.18}+\dfrac{2}{18.20}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...........+\dfrac{1}{18}-\dfrac{1}{20}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{20}\right)\)
\(=2.\dfrac{502}{1005}=\dfrac{1004}{1005}\)
\(\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{16.18}+\dfrac{4}{18.20}\)
= \(\dfrac{4}{2}\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{16.18}+\dfrac{2}{18.20}\right)\)
= \(\dfrac{4}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{16}-\dfrac{1}{18}+\dfrac{1}{18}-\dfrac{1}{20}\right)\)
= \(\dfrac{4}{2}\left(\dfrac{1}{2}-\dfrac{1}{20}\right)\)
= \(\dfrac{4}{2}.\dfrac{9}{20}\)
= \(\dfrac{9}{10}\)
