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.

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.