What is the average of the first five positive even numbers that are divisible by 9?

Introduction / Context:
This question asks for the average of a specific finite sequence of numbers: the first five positive even numbers that are also divisible by 9. It tests understanding of multiples, even numbers and the formula for the mean of a list of numbers. The structure of the sequence makes the calculations simple once the correct numbers are identified.

Given Data / Assumptions:

• We consider only positive even numbers.
• The numbers must be divisible by 9.
• We need the first five such numbers.
• We then compute their arithmetic mean.

Concept / Approach:
A number divisible by 9 is a multiple of 9. To also be even, it must be a multiple of 18, since 18 is the least common multiple of 2 and 9. Therefore, the required numbers form an arithmetic progression with first term 18 and common difference 18. Once we list the first five terms, we can either sum them and divide by 5 or use the property that the mean of an arithmetic progression equals the average of the first and last terms.

Step-by-Step Solution:
The smallest positive multiple of 18 is 18. The first five positive even numbers divisible by 9 are: 18, 36, 54, 72 and 90. Confirm each is divisible by 9: 18 / 9 = 2, 36 / 9 = 4, 54 / 9 = 6, 72 / 9 = 8, 90 / 9 = 10. Now compute the sum: 18 + 36 + 54 + 72 + 90. 18 + 36 = 54, so the sum becomes 54 + 54 + 72 + 90. 54 + 54 = 108, and 72 + 90 = 162. Total sum = 108 + 162 = 270. Average = sum / number of terms = 270 / 5 = 54.

Verification / Alternative check:
Since the numbers form an arithmetic progression with first term 18 and last term 90, the mean of the sequence can also be found as (first term + last term) / 2 = (18 + 90) / 2 = 108 / 2 = 54. This matches the result obtained by direct summation and division, confirming the calculation.

Why Other Options Are Wrong:
Options such as 50, 48, 60 or 62 do not correspond to the mean of the correctly identified sequence. These values might arise from mistakes like picking wrong multiples of 9, forgetting that the numbers must be even, or miscomputing the sum or the division. Only 54 is consistent with both the definition of the sequence and the arithmetic mean formula.

Common Pitfalls:
A common error is to list multiples of 9 that are not even, or to list the first five even numbers and then check divisibility by 9 incorrectly. Another mistake is to misapply the mean formula, for example dividing by an incorrect number of terms. Recognising that the sequence is made of multiples of 18 and using the arithmetic progression approach makes this problem quick and reliable to solve.

Final Answer:
The average of the first five positive even numbers divisible by 9 is 54.