The average of 6 consecutive integers is 15/2 (that is, 7.5). If these six integers are in increasing order, what is the average of the last three integers in this sequence?

Introduction / Context:
This question involves consecutive integers and their averages. We are given the average of 6 consecutive integers and asked to find the average of the last three numbers. Consecutive integers form an arithmetic progression with common difference 1, which allows us to represent them algebraically and compute required averages quickly.

Given Data / Assumptions:

• There are 6 consecutive integers.
• The average of these 6 integers is 15/2 = 7.5.
• We need the average of the last three integers in this ordered list.

Concept / Approach:
Let the six consecutive integers be represented in terms of a variable. Because they are consecutive, they can be written as n, n + 1, n + 2, n + 3, n + 4, n + 5. Their average will be the middle value between the third and fourth integer, which is n + 2.5. Setting this equal to the given average allows us to solve for n and thereby obtain all six integers. Then we directly compute the average of the last three integers.

Step-by-Step Solution:
Let the six consecutive integers be: n, n + 1, n + 2, n + 3, n + 4, n + 5. Average of these integers = (n + (n + 1) + (n + 2) + (n + 3) + (n + 4) + (n + 5)) / 6. Sum = 6n + 15, so average = (6n + 15) / 6 = n + 2.5. Given average = 15/2 = 7.5. So, n + 2.5 = 7.5. n = 7.5 - 2.5 = 5. Therefore, the six integers are: 5, 6, 7, 8, 9, 10. The last three integers are: 8, 9, 10. Average of last three integers = (8 + 9 + 10) / 3 = 27 / 3 = 9.
Verification / Alternative check:
Check the original average: (5 + 6 + 7 + 8 + 9 + 10) / 6. Sum = 45, average = 45 / 6 = 7.5 = 15/2, which confirms our list of integers. Thus, the computed average of last three integers is consistent with the given condition.
Why Other Options Are Wrong:
Option A (8): This would be the middle of 7, 8, 9 and not of 8, 9, 10. Option C (17/2 = 8.5): Does not match the average of 8, 9 and 10. Option D (7): Too small and corresponds to the lower part of the sequence. Option B (9): Correct, as it equals the average of the last three integers 8, 9 and 10.
Common Pitfalls:
Some students mistakenly think the given average is one of the integers in the sequence, which is not true here because the average is 7.5, a non integer. Others may forget that six numbers will not have a single middle integer but rather a pair of central values. Miswriting the consecutive integers or making arithmetic errors while adding them can also lead to incorrect answers.
Final Answer:
The average of the last three integers is 9.