Question

[thu]

Tinh: 

     \(\left(1-\frac{1}{21}\right).\left(1-\frac{1}{28}\right).\left(1-\frac{1}{36}\right)...\left(1-\frac{1}{1326}\right)\)

Answer 1

[Đức Phạm]

\(\left(1-\frac{1}{21}\right)\cdot\left(1-\frac{1}{28}\right)\cdot\left(1-\frac{1}{36}\right)\cdot...\cdot\left(1-\frac{1}{1326}\right)\)

\(=\frac{20}{21}\cdot\frac{27}{28}\cdot\frac{35}{36}\cdot...\cdot\frac{1325}{1326}\)

\(=\frac{40}{42}\cdot\frac{54}{56}\cdot\frac{70}{72}\cdot...\cdot\frac{2650}{2652}\)

\(=\frac{5.8}{6.7}\cdot\frac{6.9}{7.8}\cdot\frac{7.10}{8.9}\cdot....\cdot\frac{50.53}{51.52}\)

\(=\frac{5\cdot6\cdot7\cdot.....\cdot50}{6\cdot7\cdot8\cdot....\cdot51}\cdot\frac{8\cdot9\cdot10\cdot....\cdot53}{7\cdot8\cdot9\cdot...\cdot52}=\frac{5}{51}\cdot\frac{53}{7}=\frac{265}{357}\)

Answer 2

[NgườiBíẨn2008]

\(\left(1-\frac{1}{21}\right)\left(1-\frac{1}{28}\right)\left(1-\frac{1}{36}\right)\cdot\cdot\cdot\cdot\cdot\left(1-\frac{1}{1326}\right)\)

\(=\frac{20}{21}\cdot\frac{27}{28}\cdot\frac{35}{36}\cdot\cdot\cdot\cdot\cdot\frac{1325}{1326}\)

\(=\frac{40}{42}\cdot\frac{54}{56}\cdot\frac{70}{72}\cdot\cdot\cdot\cdot\cdot\cdot\frac{2650}{2652}\)

\(=\frac{5\cdot8}{6\cdot7}\cdot\frac{6\cdot9}{7\cdot8}\cdot\frac{7\cdot10}{8\cdot9}\cdot\cdot\cdot\cdot\cdot\frac{50\cdot53}{51\cdot52}\)

\(=\frac{\left(5\cdot8\right)\left(6\cdot9\right)\left(7\cdot10\right)\cdot\cdot\cdot\cdot\cdot\left(50\cdot53\right)}{\left(6\cdot7\right)\left(7\cdot8\right)\left(8\cdot9\right).....\left(51\cdot52\right)}\)

\(=\frac{\left(5\cdot6\cdot7\cdot\cdot\cdot\cdot\cdot50\right)\cdot\left(8\cdot9\cdot10\cdot\cdot\cdot\cdot\cdot53\right)}{\left(6\cdot7\cdot8\cdot\cdot\cdot\cdot\cdot51\right)\cdot\left(7\cdot8\cdot9\cdot\cdot\cdot\cdot\cdot52\right)}\)

\(=\frac{5\cdot53}{51\cdot7}=\frac{265}{357}\)