Bài 3:
a: \(D=\dfrac{1}{3}+\dfrac{1}{6}+...+\dfrac{1}{36}+\dfrac{1}{45}\)
\(=\dfrac{2}{6}+\dfrac{2}{12}+...+\dfrac{2}{72}+\dfrac{2}{90}\)
\(=2\left(\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{72}+\dfrac{1}{90}\right)\)
\(=2\left(\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{8\cdot9}+\dfrac{1}{9\cdot10}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{8}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{10}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{10}\right)=2\cdot\dfrac{5-1}{10}=\dfrac{1}{5}\cdot4=\dfrac{4}{5}\)
b: \(E=\dfrac{2}{20}+\dfrac{2}{30}+...+\dfrac{2}{210}+\dfrac{2}{240}\)
\(=2\left(\dfrac{1}{20}+\dfrac{1}{30}+...+\dfrac{1}{210}+\dfrac{1}{240}\right)\)
\(=2\left(\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+...+\dfrac{1}{14\cdot15}+\dfrac{1}{15\cdot16}\right)\)
\(=2\left(\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{14}-\dfrac{1}{15}+\dfrac{1}{15}-\dfrac{1}{16}\right)\)
\(=2\left(\dfrac{1}{4}-\dfrac{1}{16}\right)=2\cdot\dfrac{4-1}{16}=\dfrac{3}{8}\)
