How to Find the Sum of Integers on the GMAT

In the quant section of the GMAT, there are a fair number of formulae to know in order to answer the ensemble of questions that may be asked of you. Most of them are covered in any basic test prep material, but a formula is always just a short hand version of a much longer manual process.

There is an anecdote about a primary school teacher who wanted to keep a misbehaved child busy for a period, so she asked him to sum up all the numbers from 1 to 100. To her dismay, the child answered the question in a matter of seconds, and the answer was correct. The child explained to his teacher that, instead of simply adding 1+2+3…, you could create a pairwise addition that would always yield the same number. If you added 1 to 100, you would get 101. If you added 2 to 99, you would still get 101. If you added 3 to 98, you’d still get 101, and so on. Thus the addition of 100 different numbers could be turned into a multiplication of two simple numbers: 101 x 50. The student in question was mathematical prodigy Carl Friedrich Gauss.

Now such brilliance is hard to see the first time, but easy to transform into a formula once it has been discovered.  Why did this shortcut method work? Quite simply, because the numbers are in arithmetic progression, which is to say, the spacing between each number is a constant. This allows the mean to be equal to the median, and the median is a very easy number to calculate. In an odd numbered set, it is the middle term. In an even numbered set, it is the average of the two middle terms. Thus, from 1 to 100, the median is the average of 50 and 51, or 50.5. Multiplying this number by the number of terms gives us 50.5 x 100, or 5,050. Had we asked for the sum of all numbers from 0 to 100, we’d have a median (and average) of exactly 50, and 101 terms. This is the same equation that Gauss used, and obviously yields the same result as 1 to 100 since we’re only adding the term 0.

In general, the formula is going to be the Mean x Number of terms. There are two possible caveats with this formula, the first is in calculating the number of terms quickly, and the second is in taking into account calculations where the frequency might come into play. Let’s demonstrate this with a (real life) GMAT question:

What is the sum of all even integers from 650 to 750, inclusive?

3,500

35,000

35,700

70,000

70,700

This question can appear daunting if you don’t know how to approach such problems, so let’s delve into the mathematics of how to solve it (without taking a wild Gauss).

First, let’s consider what would happen if we dropped the word “even” from the question. We’d want all the integers from 650 to 750. In this arithmetic progression, the median would be 700, and thus so would the mean. The number of terms is easiest to consider as (Biggest – Smallest) + 1. This is because we have to account for both end points. Consider the number of terms from 50 to 60. We’d have 11 terms (you may want to count using fingers and toes). The total would thus be Mean (700) x number of terms (101), which would equal 70,700. This is the trap answer E because we discounted the word “even”, and clearly 70,000 is the same trap but without the +1 extra term (650 to 749, in effect). It thus has to be either A, B or C, but which one?

The issue of missing terms is addressed by simply dividing by the frequency. In this case, even numbers account for ½ of the total numbers, so we’ll have to divide by 2. The only other issue is to determine where the endpoints must lie. 650 is even, and so is 750, so we don’t need to change anything except the frequency, which becomes: Mean (700) x number of terms ((100/2) + 1). The extra term always needs to be added back in, so in effect we have 700 x 51, or 35,700.

The correct answer is C. 

B is the trap answer in case we overlooked the +1 at the end.

For completion’s sake, let’s examine what would happen had the question asked us for only the odd numbers. The same average would still exist, but the number of terms would now be (biggest-smallest/2) +1, which is ((749-651)/2) +1 = (98/2) +1 = 49+1, or exactly 50. Thus the sum of all the odd numbers from 650 to 750 is 35,000. This jives with our calculations of the even numbers summing to 35,700 and the total of all numbers being 70,700.

Given an infinite amount of time and a calculator (or abacus), you could easily find the answers to these questions without a general case formula. However, since you have about 2 minutes to read the question, accomplish all required calculations and lock in an answer choice, you are best served to have the formulae memorized or a strong notion of how to calculate the answer using the information provided. The GMAT is an exam about how you think more than what you know, but you don’t have to be Gauss if you’re well prepared for questions you expect to see.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you occasional tips and tricks for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.

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