Question

( 1 - 1/2 ) . ( 1 - 1/3) . ( 1 - 1/4) ... ( 1 - 1/2009) . ( 1 - 1/2010)

Answer 1

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

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

\(=\dfrac{1.2.3...2009}{2.3.4...2010}\)

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

Answer 2

\(\left(1-\dfrac{1}{2}\right)\cdot\left(1-\dfrac{1}{3}\right)\cdot\left(1-\dfrac{1}{4}\right)\dots\left(1-\dfrac{1}{2009}\right)\cdot\left(1-\dfrac{1}{2010}\right)\)

\(=\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot\dfrac{3}{4}\cdot\cdot\cdot\dfrac{2008}{2009}\cdot\dfrac{2009}{2010}\)

\(=\dfrac{1\cdot2\cdot3\dots2008\cdot2009}{2\cdot3\cdot4\dots2009\cdot2010}\)

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