SOLUTION: the product (1- 1/2)(1- 1/3)(1- 1/4) (1-1/5)…(1-1/99)(1-1/100) is equal to
a) 1/99 b) 1/80 c) 1/90 d) 1/100 e) none of these
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Question 198372: the product (1- 1/2)(1- 1/3)(1- 1/4) (1-1/5)…(1-1/99)(1-1/100) is equal to
a) 1/99 b) 1/80 c) 1/90 d) 1/100 e) none of these
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Take note that we can combine the terms of
\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
\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:
So this means that
\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})
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