Multiplication of rational numbers

Question

Find the product of:

1/2 x 2/3 x 3/4 x 4/5 x ......x 99/100 .

 

What is the product when the last rational number is n-1/n?

Answer 1

If you just set up say the first 4 terms, you'll notice everything cancels out except 1 at the beginning, and /5 at the end, which gives 1/5 as the answer.

 

So if the last term is 99/100, then the final answer would be 1/100

 

So if the last rational number is n-1/n, the answer would be 1/n

Answer 2

If you write this out this way...

 

 1      2      3

--- x --- x ---   etc.

 2      3      4

 

...you can see that the denominator of one fraction cancels the numerator of the next one.  All you're left with is 1 in the numerator from the first term, and 100 in the denominator from the last term, since there's no next fraction after that to cancel the 100.  So the result is just 1/100.

 

And you can see that the same pattern will occur for any sequence like this.  There will always be 1 left in the numerator, and the only thing left in the denominator will be the last term's denominator.  Therefore, if the last term is n-1/n, the product will be 1/n.