Question

N= (1- 1/3)+(1-1/6)+(1-1/10)+(1-1/15)+...+(1-1/780)

Answer 1

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

\(\Rightarrow N=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}......\frac{779}{780}=\frac{4}{6}.\frac{10}{12}.\frac{18}{20}.....\frac{1558}{1560}\)

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

\(\Rightarrow N=\frac{\left(1.2.3.....38\right)\left(4.5.6.....41\right)}{\left(2.3.4.....39\right)\left(3.4.5.....40\right)}=\frac{1.41}{39.3}=\frac{41}{117}\)