The following three properties apply to all evenly spaced sequences. However, just because a sequence has these properties does not necessarily mean that the sequence is evenly spaced.
The arithmetic mean (average) and median are equal to each other. In other words, the average of the elements in the set can be found by figuring out the median, or “middle number.”
What is the arithmetic mean of 4, 8, 12, 16, and 20?
In this example there are five consecutive multiples of 4. The median is the third largest, or 12. Since this is an evenly spaced set, the arithmetic mean (average) is also 12.
What is the arithmetic mean of 4, 8, 12, 16, 20, and 24?
In this example there are six consecutive multiples of 4. The median is the arithmetic mean (average) of the third largest and fourth largest, or the average of 12 and 16. Thus, the median is 14. Because this is an evenly spaced sequence, the average is also 14.
The mean and median of a sequence are equal to the average of the first and last terms.
What is the arithmetic mean of 4, 8, 12, 16, and 20?
In this example, 20 is the largest (last) number and 4 is the smallest (first). The arithmetic mean and median are therefore equal to (20 + 4) ÷ 2 which is 12.
What is the arithmetic mean of 4, 8, 12, 16, 20, and 24?
In this example, 24 is the largest (last) number and 4 is the smallest (first). The arithmetic mean and median are therefore equal to (24 + 4) ÷ 2 which is 14.
Thus, for all evenly spaced sequences, just remember: the average equals (First + Last) ÷ 2.
The sum of the elements in the sequence equals the arithmetic mean (average) times the number of items in the sequence.
This property applies to all sequences, but it takes on special significance in the case of evenly spaced sequences because the “average” is not only the arithmetic mean, but also the median.
What is the sum of 4, 8, 12, 16, and 20?
You have already calculated the average above; it is equal to 12. There are 5 terms, so the sum equals 12 × 5 which is 60.
What is the sum of 4, 8, 12, 16, 20, and 24?
You have already calculated the average above; it is equal to 14. There are six terms, so the sum equals 14 × 6 which is 84.
What is the sum of the numbers 13, 14, 15, and 16?
If x = 3, what is the sum of 2x, (2x + 1), (2x + 2), (2x + 3), and (2x + 4)?