Question 198372
Take note that we can combine the terms of 


*[Tex \LARGE \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)\ldots\left(1-\frac{1}{99}\right)\left(1-\frac{1}{100\right)]



to get 


*[Tex \LARGE \left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)\ldots\left(\frac{98}{99}\right)\left(\frac{99}{100\right)]



Now take note that the second numerator (2) cancels out with the first denominator (2). Likewise, the third numerator (3) cancels out with the second denominator (3). Furthermore, the fourth numerator (4) cancels out with the third denominator (4). This pattern continues on indefinitely. So after these terms cancel out, we're simply left with:


*[Tex \LARGE \frac{1}{100}]



So this means that 



*[Tex \LARGE \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)\ldots\left(1-\frac{1}{99}\right)\left(1-\frac{1}{100\right) = \frac{1}{100}]