Question

\(\left(1-\frac{1}{2}\right).\left(1-\frac{1}{3}\right).\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{2010}\right)\)

Answer 1

\(\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)...\left(1-\frac{1}{2010}\right)\)

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2009}{2010}\)

\(=\frac{1.2.3.4.5....2008.2009}{2.3.4....2009.2010}\)

\(=\frac{1}{2010}\)

Answer 2

\(\left(1-\frac{1}{2}\right).\left(1-\frac{1}{3}\right).\left(1-\frac{1}{4}\right).....\left(1-\frac{1}{2010}\right)\)

\(=\left(\frac{2}{2}-\frac{1}{2}\right).\left(\frac{3}{3}-\frac{1}{3}\right).\left(\frac{4}{4}-\frac{1}{4}\right).....\left(\frac{2010}{2010}-\frac{1}{2010}\right)\)

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{2009}{2010}=\frac{1.2.3....2009}{2.3.4....2010}=\frac{1}{2010}\)