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Source: Official Guide for GMAT Review 2016 Problem Solving; #100


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?

4 Explanations


Alternatively to improve understanding, we could think in terms of decimals as well.

The numbers of integers of M is (300 - 201) + 1 = 100
The sum of M starts at 0.005, while the remaining integers decrease until only 0.003333 is added.

Consider the following:
1/200 = 0.005
1/225 = 0.004444...
1/250 = 0.004
1/275 = 0.003636...
1/300 = 0.003333... (think of 1/3)

Now the sum includes less than one hundred 0.005's.
If we were given 100 0.005's we would have a total sum of 100*0.005 = 0.5 (= 1/2)

We receive more than one hundred 0.003333...'s
If we were given one hundred 0.0033...'s we would have a total sum of 100*0.003333... = 0.3333 ( = 1/3)

The sum of M has to be smaller than 0.5, but greater than 0.3333...
(1/3) < M < (1/2)

Oct 11, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Thanks for sharing this alternative way of looking at the problem! Happy studying :)

Oct 14, 2016 • Reply


Gravatar Cydney Seigerman, Magoosh Tutor

Hi Tayabur,

Good question! The reason we multiply by 100 each time is because we're considering two hypothetical situations:

1. All of the terms = 1/200
2. All of the terms = 1/300

This is to help us find the limits of M. Since there are 100 terms from 1/201 to 1/300, there would be 100 terms in both situations above. So, to find the sum if all 100 terms were equal to 1/200, we would add 1/200 100 times, which is equivalent to finding the product 100*(1/200). The same process could be followed replacing 1/200 with 1/300 in the second situation. :)

I hope this helps :D

Mar 23, 2016 • Comment


JESUS J Tellez

If one misses the state there a relatively simple algebraic solution knowing it's an evenly spaced set? Average of first and last terms to equal average of set * # of items of set...?

Mar 10, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Hi Jesus :)

It turns out that the set of number is not evenly spaced. While the consecutive integers 201 to 300 are evenly spaced, their reciprocals are not. Let's look at the first three terms to see what I mean:

1/202 - 1/201 = approx -0.000024629
1/203 - 1/202 = approx -0.000024387

While the difference between the first and second and second and third terms is very close, it is not exactly the same. For that reason, we cannot use the assumption that the terms in the sequence are evenly spaced to find the sum of these terms.

With that in mind, the explanations given in our explanation are generally the fastest and simplest methods there are, and I recommend that you work to understand the solution Mike explains :)

I hope this helps a little!

Mar 14, 2016 • Reply


Gravatar Mike McGarry, Magoosh Tutor

Aug 14, 2015 • Comment

Anika Mozammel

why are we multiplying 100 each time?

Mar 18, 2016 • Reply

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