Question

[Trần Lê Thùy Lịn]

1 + 2/3 + 5/6 + 9/10 + 20/21 + 27/28 + 35/36 + 44/45

Answer 1

[Nguyễn Lương Bảo Tiên]

Dấu chấm là dấu nhân

\(1+\frac{2}{3}+\frac{5}{6}+\frac{9}{10}+...+\frac{44}{45}\)

\(=1+1-\frac{1}{3}+1-\frac{1}{6}+1-\frac{1}{10}+...+1-\frac{1}{45}\)

\(=8-2\cdot\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{9\cdot10}\right)\)

\(=8-2\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\right)\)

\(=8-2\cdot\left(\frac{1}{2}-\frac{1}{10}\right)\)

\(=8-2\cdot\frac{2}{5}\)

\(=8-\frac{4}{5}\)

\(=\frac{36}{5}\)

Answer 2

[Đỗ Ngọc Hải]

Đặt \(A=1+\frac{2}{3}+\frac{5}{6}+\frac{9}{10}+\frac{20}{21}+\frac{27}{28}+\frac{35}{36}+\frac{44}{45}\)

\(=1+1-\frac{1}{3}+1-\frac{1}{6}+1-\frac{1}{10}+1-\frac{1}{21}+1-\frac{1}{28}+1-\frac{1}{36}+1-\frac{1}{45}\)

\(=8-\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}\right)\)

\(\Leftrightarrow\frac{1}{2}.A=4-\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+\frac{1}{9.10}\right)\)

\(=4-\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}\right)\)

\(=4-\left(\frac{1}{2}-\frac{1}{5}+\frac{1}{6}-\frac{1}{10}\right)\)

\(=4-\frac{11}{30}=\frac{109}{30}\)

\(\Rightarrow A=\frac{109}{15}\)