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Hints and solutions to
Try a Simpler Version #2
Find the product:
( 1 - 1/2)(1 - 1/3)(1 - 1/4)......(1 - 1/98)(1 -
1/99)(1 - 1/100) = ?
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This one looks like it's going to be hard, but there is a pattern
to be noticed! Let's simplify the problem by taking fewer factors. We'll
always start with 1 - 1/2, but simply not go as far:
- 1 - 1/2 = 1/2
- (1 - 1/2)(1 - 1/3) = (1/2)(2/3) = 1/3
- (1 - 1/2)(1 - 1/3)(1 - 1/4) = (1/2)(2/3)(3/4) = 1/4
- and so on...
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By doing the calculations for this one by hand, you can see the
pattern of cancellations -- all the numerators except the 1 at the
beginning cancel with all of the denominators except the very last one. So
you can see why the product all the way up to (1-1/100) will be 1/100.
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If you used a calculator, you probably didn't see the
cancellation
right away.
But you can notice that the first few answers are 0.5, 0.33333.., 0.25,
0.2, and hopefully you recognize these numbers as 1/2, 1/3, 1/4, 1/5, etc.
It's possible then to predict that the product all the way up to (1-1/100)
will be
0.01 = 1/100. And the "cancellation method" provides the proof.
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