Question

Tính:

\(A=\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{6}\right)\left(1-\dfrac{1}{10}\right)...\left(1-\dfrac{1}{780}\right)\)

Answer 1

\(A=\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{6}\right)\left(1-\dfrac{1}{10}\right)......\left(1-\dfrac{1}{780}\right)\)

= \(\dfrac{2}{3}.\dfrac{5}{6}.\dfrac{9}{10}.......\dfrac{779}{780}\)

= \(\dfrac{4}{6}.\dfrac{10}{12}.\dfrac{18}{20}.....\dfrac{1558}{1560}\)

= \(\dfrac{1.4}{2.3}.\dfrac{2.5}{3.4}.\dfrac{3.6}{4.5}......\dfrac{38.41}{39.40}\)

= \(\dfrac{1.4.2.5.3.6.....38.41}{2.3.3.4.4.5...39.40}\)

= \(\dfrac{\left(1.2.3....38\right)\left(4.5.6....41\right)}{\left(2.3.4....39\right)\left(3.4.5...40\right)}\)

= \(\dfrac{1}{39}.\dfrac{41}{3}\) = \(\dfrac{41}{117}\)