M is the sum of the reciprocals

M is the sum of the reciprocals of the consecutive integers from 201 to 300 inclusive. Which of the following is true?

3 Explanations

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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?

Nov 14, 2016 • Comment

Sam Kinsman

Hi Tobias,

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:

1+2+3+4+5 = 15

3 * 5 = 15

However, this doesn't work with fractions.

Nov 15, 2016 • Reply

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Number of fractions in the secuence:

last - first + 1 = 300 - 201 + 1 = 301 - 201 = 100

So, we have 50 couples of fractions (The sum of the first term and the last one is equal to the sum of the second term and the term #99, and so on)

sum of the first and the last term:

(1/201) + (1/300)

Let´s simplify (we will obtain a result slightly greater than the "real" one, since 1/200 is greater than 1/201.

(1/200) + (1/300) = 5 / 600

The sum of the sequence will be this term multiply by the # of couples (every couple sum 5/600 aprox).

(5/600) * # couples = (5/600) * 50 = (250/600) = 5/12

Since our term is slightly greater than the real one, the result will be slightly less than 0,4.

Option A links to our prediction.

0.3333 < M < 0.5

pdta: thanks for pointing out 5/600 :)

Jan 25, 2015 • Comment

How does (1/200) + (1/300) = 6 / 600? Is it not 5/600 instead?

Mar 24, 2015 • Reply

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Mike McGarry, Magoosh Tutor

Dec 27, 2013 • Comment

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M is the sum of the reciprocals

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