\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)\cdot\cdot\cdot\left(1-\dfrac{1}{100^2}\right)\\ =\left(1-\dfrac{1}{2}\right)\left(1+\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1+\dfrac{1}{3}\right)\cdot\cdot\cdot\left(1-\dfrac{1}{100}\right)\left(1+\dfrac{1}{100}\right)\\ =\dfrac{1}{2}\cdot\dfrac{3}{2}\cdot\dfrac{2}{3}\cdot...\cdot\dfrac{99}{100}\cdot\dfrac{101}{100}\\ =\left(\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot...\cdot\dfrac{99}{100}\right)\left(\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot...\cdot\dfrac{101}{100}\right)\\ =\dfrac{1\cdot2\cdot...\cdot99}{2\cdot3\cdot...\cdot100}\cdot\dfrac{3\cdot4\cdot...\cdot101}{2\cdot3\cdot...\cdot100}\)
`=(1*\cancel(2)*...*\cancel(99))/(\cancel(2)*\cancel(3)*...*100)*(\cancel(3)*\cancel(4)*...*101)/(2*\cancel(3)*...*\cancel(100))`
`=1/100*101/2`
`=101/200`
