You may recall that the average of a set of consecutive integers is the middle number (the middle number of any dataset is always its median—more on this later). This is true for any set in which the terms are spaced evenly apart. For example:
The average of the set {3, 5, 7, 9, 11} is the middle term 7, because all the terms in the set are spaced evenly apart (in this case, they are spaced two units apart).
The average of the set {12, 20, 28, 36, 44, 52, 60, 68, 76} is the middle term 44, because all the terms in the set are spaced evenly apart (in this case, they are spaced eight units apart).
Note that if an evenly spaced set has two “middle” numbers, the average of the set is the average of these two middle numbers. For example:
The average of the set {5, 10, 15, 20, 25, 30} is 17.5, because this is the average of the two middle numbers: 15 and 20.
You do not have to write out each term of an evenly spaced set to find the middle number—the average term. All you need to do to find the middle number is to add the first and last terms and divide that sum by 2. For example:
The average of the set {101, 111, 121 … 581, 591, 601} is equal to 351, which is the sum of the first and last terms (101 + 601 = 702) divided by 2. This approach is especially attractive if the number of terms is large.
What is the average of the set {2, 5, 8, 11, 14}?
What is the average of the set {−1, 3, 7, 11, 15, 19, 23, 27}?