M is the sum of the reciprocals of the consecutive integers from 201 to 300 inclusive. Which of the following is true?
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Tobias Meyerding
wouldn't it be faster to simply take the middle value of this list (1/250) and multiply by 100 (number of different values) which yields 100/250 = 2/5; a value clearly between 1/2 and 1/3?
That's a clever way of looking at it! However, it doesn't quite work. If we take that approach, we happen to get a similar value to M. But that's actually a bit of a coincidence. Let me explain why this doesn't work using a simpler example!
Let's say we have the following sum:
1/1 + 1/2 + 1/3 + 1/4 + 1/5.
If we solve this with a calculator, we get 2.283
If we use the approach you suggested for this one, we would have the middle value (1/3) multiplied by 5.
(1/3) * 5 = 5/3 = 1.66
This is not the same as 2.283. So the method doesn't really work when we have fractions.
If we have a sum that doesn't involve fractions, the method does work. For example:
3 Explanations