Question

(1-1/3)*(1-1/6)*(1-1/10)*(1-1/15)*...*(1-780)

Answer 1

\(\left(1-\frac{1}{3}\right).\left(1-\frac{1}{6}\right).\left(1-\frac{1}{10}\right).\left(1-\frac{1}{15}\right)...\left(1-\frac{1}{780}\right)\)

\(=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}.\frac{14}{15}...\frac{779}{780}\)

\(=\frac{4}{6}.\frac{10}{12}.\frac{18}{20}.\frac{28}{30}...\frac{1558}{1560}\)

\(=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}.\frac{4.7}{5.6}...\frac{38.41}{39.40}\)

\(=\frac{1.2.3.4...38}{2.34.5...39}.\frac{4.5.6.7...41}{3.4.5.6...40}\)

\(=\frac{1}{39}.\frac{41}{3}=\frac{41}{297}\)

Answer 2

41/297