Dấu chấm là dấu nhân
\(1+\frac{2}{3}+\frac{5}{6}+\frac{9}{10}+...+\frac{44}{45}\)
\(=1+1-\frac{1}{3}+1-\frac{1}{6}+1-\frac{1}{10}+...+1-\frac{1}{45}\)
\(=8-2\cdot\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{9\cdot10}\right)\)
\(=8-2\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\right)\)
\(=8-2\cdot\left(\frac{1}{2}-\frac{1}{10}\right)\)
\(=8-2\cdot\frac{2}{5}\)
\(=8-\frac{4}{5}\)
\(=\frac{36}{5}\)
Đặt \(A=1+\frac{2}{3}+\frac{5}{6}+\frac{9}{10}+\frac{20}{21}+\frac{27}{28}+\frac{35}{36}+\frac{44}{45}\)
\(=1+1-\frac{1}{3}+1-\frac{1}{6}+1-\frac{1}{10}+1-\frac{1}{21}+1-\frac{1}{28}+1-\frac{1}{36}+1-\frac{1}{45}\)
\(=8-\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}\right)\)
\(\Leftrightarrow\frac{1}{2}.A=4-\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+\frac{1}{9.10}\right)\)
\(=4-\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}\right)\)
\(=4-\left(\frac{1}{2}-\frac{1}{5}+\frac{1}{6}-\frac{1}{10}\right)\)
\(=4-\frac{11}{30}=\frac{109}{30}\)
\(\Rightarrow A=\frac{109}{15}\)
