Sequences: Consecutive Integers - Average=Median

This example shows the most common occurrence of this property, which is sets of consecutive integers. However, this property is not confined to consecutive integers alone; In any set of integers which constitute an arithmetic sequence - any set of integers with a constant difference between each consecutive terms (be it a difference of 1 (1,2,3), two (2, 4 ,6) or 5 (2, 7, 12)) - the average = the median. Think of the average of a set of consecutive numbers as the exact "middle" of the set.

Your wish is our command.

A whale goes on a feeding frenzy that lasts for 9 hours. For the first hour, he catches and eats x kilograms of plankton. In every hour after the first, he consumes 3 kilograms of plankton more than he consumed in the previous hour. If, by the end of the frenzy, the whale consumes a whopping 450 kilograms of plankton, how many kilograms did he consume on the sixth hour?

Remember:

[[summary]]

Incorrect.

38 is, in fact, the value of x, i.e. the number of kilos of Plankton that the whale consumes in the first hour. But what did the question ask?

Incorrect.

50 is, in fact, the value of the median, i.e. the number of kilos of Plankton that the whale consumes in the fifth hour. But what did the question ask?

Incorrect.

Incorrect.

A whale goes on a feeding frenzy that lasts for 9 hours. For the first hour, he catches and eats x kilograms of plankton. In every hour after the first, he consumes 3 kilograms of plankton more than he consumed in the previous hour. If, by the end of the frenzy, the whale consumes a whopping 450 kilograms of plankton, how many kilograms did he consume on the sixth hour?

A seemingly tough question, requiring several steps.

Fortunately, this question, and others like it, can be solved with the use of the properties of arithmetic sequences. An arithmetic sequence has several properties which are tested in GMAT problems. One of them is the average property:

in a set of integers with a constant difference between them (including consecutive integers, where the difference is 1): Average of the set = Median of the set. 

Take a look at this simple set of consecutive integers: {1, 2, 3}.

The median of the set is easily recognizable - since the set has an odd number of terms, the median is the number in the middle, or 2.

The Average of the set is total / # of items = (1+2+3) / 3 = 6/3 = 2.

This stays true even if the set includes an even number of consecutive integers, as in the set {1, 2, 3, 4}. since there is an even number of members in the set, the median is calculated as the average of the two middle terms: (2+3) / 2 = 5 / 2 = 2.5.

Surprisingly enough, the average of the set is also (1+2+3+4) / 4 = 10 / 4 = 2.5.

Correct.

The question describes an arithmetic sequence with a difference of 3: in the first hour our whale consumes x kilos, in the second (x+3), in the third (x+6), etc. Adding these together will give a total of 450, from which we can find x, but that is not an easy calculation. By the time you're done with that, you might easily forget that the question does not ask for x, but rather for the consumption in the sixth hour, which is actually x+15.

Instead, recall the average property of arithmetic sequences: Average = Median.

Since the question kindly provides the total kilos of Plankton (450) and the number of hours (9), the average hourly consumption of Plankton can be easily calculated: 450 / 9 = 50.

Therefore, the Median of our set of consecutive integers is also 50. Since the set has an odd number of members, the median is the number in the middle, or the 5th hour. If the whale consumes 50 kilos of Plankton in the 5th hour, he will consume 50+3 = 53 kilos in the sixth hour.

Quick and easy - with the right approach.