A set of four consecutive multiples of 4 has an average of 30. A second series…

Concept

When numbers are equally spaced (an arithmetic progression), their average equals the mean of the terms. For four consecutive multiples of 4 the common gap is 4, so the four terms can be written as a, a+4, a+8 and a+12, and their average is the value exactly in the middle of the set. The same idea fixes a run of consecutive even numbers once its starting value is known: each successive term is 2 more than the previous one.

Application

1. Let the four consecutive multiples of 4 be a, a+4, a+8, a+12. Their sum is 4a + 24.

2. Average = (4a + 24) / 4 = a + 6. Set a + 6 = 30, so a = 24.

3. The four multiples are therefore 24, 28, 32 and 36.

4. The second-lowest of these is 28, so the five consecutive even numbers start at 28.

5. Five consecutive even numbers from 28 are 28, 30, 32, 34, 36 (each 2 more than the last).

6. The highest term of this even series is 36.

Cross-check

Average of 24, 28, 32, 36 = 120 / 4 = 30, confirming the four multiples. Starting the even run at 28 and adding 2 four times gives 28, 30, 32, 34, 36, whose largest value is 36.