The average of 8 consecutive integers is 23/2. What is the average of the first three integers in this sequence?

Introduction / Context:
This question involves consecutive integers and averages. You are told that the average of 8 consecutive integers is 23/2, and you need to find the average of just the first three numbers in that sequence. This tests your ability to represent consecutive integers algebraically, use the mean formula and reason about symmetry in evenly spaced sequences.

Given Data / Assumptions:

- There are 8 consecutive integers. - Their average is 23/2. - The integers are equally spaced with a difference of 1. - We want the average of the first three integers in this sequence.

Concept / Approach:
Consecutive integers can be represented as k, k + 1, k + 2, ..., k + 7. For consecutive terms, the average is simply the mid point, which is equal to the average of the first and last terms. Once we determine the first integer k using the given overall average, we can identify the first three integers and compute their average directly. This approach avoids finding all eight values individually by focusing on the pattern and properties of arithmetic sequences.

Step-by-Step Solution:
Step 1: Let the 8 consecutive integers be k, k + 1, k + 2, k + 3, k + 4, k + 5, k + 6 and k + 7. Step 2: The average of these 8 integers is (first + last) / 2 = (k + (k + 7)) / 2 = (2k + 7) / 2. Step 3: We are told that this average equals 23/2. Step 4: Set up the equation (2k + 7) / 2 = 23 / 2. Step 5: Since denominators are the same and non zero, equate the numerators: 2k + 7 = 23. Step 6: Solve for k: 2k = 23 - 7 = 16, so k = 8. Step 7: Therefore the 8 consecutive integers are 8, 9, 10, 11, 12, 13, 14 and 15. Step 8: The first three integers are 8, 9 and 10. Step 9: Compute their average: (8 + 9 + 10) / 3 = 27 / 3 = 9.

Verification / Alternative check:
We can verify by checking the given average. The sum of all 8 integers is 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 92. The average is then 92 / 8 = 11.5, which is 23/2, matching the condition. Since the first three numbers are definitely 8, 9 and 10, their average must be 9, confirming our answer.

Why Other Options Are Wrong:
Option B, 19/2, would correspond to 9.5, which is not the average of 8, 9 and 10. Option C, 8, is the first integer but not the average of the first three. Option D, 10, is the third integer but again not the average. Only 9 is consistent with the arithmetic mean of the first three terms.

Common Pitfalls:
Students sometimes confuse the given average of all 8 numbers with the average of a subset, assuming they are the same. Others may incorrectly take the average of just the first and second integers or make algebra mistakes when solving for k. Carefully setting up the representation of consecutive integers and using the formula for the mean helps avoid these issues.

Final Answer:
The average of the first three integers in the sequence is 9, which corresponds to option A.