\(G=\left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right)\left(1-\frac{1}{10}\right)....\left(1-\frac{1}{780}\right)\)
\(=\left(1-\frac{1}{\frac{2.3}{2}}\right)\left(1-\frac{1}{\frac{3.4}{2}}\right)\left(1-\frac{1}{\frac{4.5}{2}}\right).....\left(1-\frac{1}{\frac{39.40}{2}}\right)\)
Ta có : \(1-\frac{1}{\frac{n\left(n+1\right)}{2}}=\frac{n\left(n+1\right)-2}{n\left(n+1\right)}=\frac{n^2+n-2}{n\left(n+1\right)}=\frac{\left(n-1\right)\left(n+2\right)}{n\left(n+1\right)}\)
Áp dụng ta được :
\(G=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}.......\frac{38.41}{39.40}\)
\(=\frac{\left(2.3....38\right)\left(4.5.6.....41\right)}{\left(2.3.4....39\right)\left(3.4.5....40\right)}=\frac{41}{39.3}=\frac{41}{117}\)
