Find the average of all even numbers between 104 and 148 (both inclusive). These even integers form a simple arithmetic sequence. What is their average value?

Introduction / Context:
This question asks for the average of even numbers within a given inclusive range. Since consecutive even numbers differ by 2, they form an arithmetic sequence. For such a sequence, we can use a shortcut to find the average rather than individually listing or summing all terms.

Given Data / Assumptions:

• Range considered: from 104 to 148 inclusive.
• Only even numbers are taken.
• 104 and 148 are both even, so they are part of the sequence.
• We need the average of all these even numbers.

Concept / Approach:
Consecutive even numbers form an arithmetic progression with common difference 2. For any arithmetic sequence, the average is equal to (first term + last term) / 2. This is true regardless of how many terms there are, as long as the spacing between terms is uniform. So we simply identify the first even number and the last even number in the range and compute their mean.

Step-by-Step Solution:
First even number in the range = 104. Last even number in the range = 148. These numbers form an arithmetic sequence: 104, 106, 108, ..., 148. Average of an arithmetic sequence = (first term + last term) / 2. Average = (104 + 148) / 2. Average = 252 / 2 = 126.
Verification / Alternative check:
Observe that the even numbers are symmetric around their average. Pairs like (104, 148), (106, 146), (108, 144) all sum to 252. Since each pair has the same sum and there are equally many such pairs, the overall average must be 252 / 2 = 126, confirming the result.
Why Other Options Are Wrong:
Option A (128): Slightly higher than the correct mean; may arise if someone miscomputes the sum. Option B (130): Too high and not equal to the midpoint of 104 and 148. Option C (124): Too low and again not matching the midpoint. Option D (126): Correct, since it is exactly the average of the first and last even numbers.
Common Pitfalls:
Some students mistakenly average 104 and 148 with 100 and 150 or perform incorrect arithmetic. Others may think they must list all even numbers and individually sum them, which is time consuming and error prone. Forgetting that the average of an arithmetic sequence is simply the mean of the first and last terms is another common oversight.
Final Answer:
The average of all even numbers between 104 and 148 inclusive is 126.