What is the average of all even numbers between 222 and 250?

Introduction / Context:
This question tests your understanding of averages for a sequence of evenly spaced numbers, in this case all even numbers in a given interval. Such sequences form an arithmetic progression, and there is a simple shortcut to find their average using only the first and last term of the sequence.

Given Data / Assumptions:

We are concerned with even numbers between 222 and 250.
All numbers are integers and differ by 2, since they are consecutive even numbers.
The term “between” here still yields the same answer whether we treat endpoints as included or excluded, due to symmetry.
We must compute the arithmetic mean of all these even numbers.

Concept / Approach:
Even numbers in order form an arithmetic progression (AP) with common difference 2. For any AP, the average of all terms is equal to (first term + last term) / 2. Therefore, once we know the smallest even number at or above 222 and the largest even number at or below 250, we can directly use this property instead of summing many terms.

Step-by-Step Solution:
Step 1: Identify the first even number in the range. 222 is itself an even number, so the smallest even number is 222. Step 2: Identify the last even number in the range. 250 is also even, so the largest even number is 250. Step 3: Use the AP average property. Average of all even numbers from 222 to 250 = (222 + 250) / 2. Compute: 222 + 250 = 472, and 472 / 2 = 236.

Verification / Alternative Check:
If we consider exclusive bounds (224 to 248), the first even number is 224 and the last is 248. Their average is (224 + 248) / 2 = 472 / 2 = 236 again. Because the even numbers are symmetrically placed around the center, both interpretations lead to the same average of 236, confirming that our result is robust.

Why Other Options Are Wrong:
232 and 234 are smaller than the true central value and would correspond to a range whose endpoints do not match 222 and 250 appropriately.
230 is even farther from the midpoint and clearly not the center of an evenly spaced set running from 222 to 250.

Common Pitfalls:
A frequent error is to assume that the average is always the middle term; this is only true when you know the exact count and list. Others may try to sum all terms manually, which is time consuming and prone to arithmetic mistakes. Remember that for any arithmetic progression, the average is simply half the sum of the first and last terms.

Final Answer:
The average of all even numbers between 222 and 250 is 236.