The average of four consecutive odd numbers is 64. What is the value of the largest of these four odd numbers?

Introduction / Context:
This problem tests understanding of arithmetic progressions and averages of consecutive odd numbers. When numbers are equally spaced, such as consecutive odd numbers, the average gives direct information about the middle of the sequence, which can be used to find each term including the largest.

Given Data / Assumptions:

• There are four consecutive odd numbers.
• The average of these four numbers is 64.
• We need to find the largest of these four odd numbers.

Concept / Approach:
Consecutive odd numbers form an arithmetic progression with common difference 2. Let the numbers be x, x + 2, x + 4 and x + 6. The average of an arithmetic progression is equal to the average of the first and last term, and is also equal to the middle of the four in this case. We can use the formula for the average to find x, then compute the largest term x + 6.

Step-by-Step Solution:
Step 1: Let the four consecutive odd numbers be x, x + 2, x + 4 and x + 6. Step 2: The average of these four numbers is given as 64. Step 3: The sum of the four numbers is x + (x + 2) + (x + 4) + (x + 6) = 4x + 12. Step 4: Average = (sum) / 4 = (4x + 12) / 4 = x + 3. Step 5: Set x + 3 equal to 64 to match the given average: x + 3 = 64. Step 6: Solve for x: x = 64 - 3 = 61. Step 7: The four numbers are 61, 63, 65 and 67. The largest is x + 6 = 61 + 6 = 67.

Verification / Alternative check:
Check the average directly from the sequence 61, 63, 65 and 67. Their sum is 61 + 63 + 65 + 67 = 256. The average is 256 / 4 = 64, which matches the given condition. Thus the sequence is correct and the largest number is confirmed as 67.

Why Other Options Are Wrong:
Values like 65, 69 or 71 cannot be the largest number in a set of four consecutive odd numbers with average 64. For example, if 69 were the largest, the sequence would be 63, 65, 67, 69, whose average is 66, not 64. Similar checks show that sequences ending in 65 or 71 do not have average 64.

Common Pitfalls:
Some learners mistakenly treat 64 as one of the odd numbers, but 64 is even and cannot be in the set. Others forget that four consecutive odd numbers around 64 must lie symmetrically, with two below and two above the average. The algebraic approach or simple reasoning about symmetry both lead quickly to the correct answer.

Final Answer:
The largest of the four consecutive odd numbers is 67.