The average of four consecutive odd natural numbers is 64. These four odd numbers follow one after another in increasing order. What is the value of the largest of these four odd numbers?

Introduction / Context:
This problem tests understanding of averages and sequences of consecutive odd numbers. Consecutive odd numbers form an arithmetic progression with a common difference of 2. Knowing the average allows us to find the middle of this progression and then determine all the numbers in the set, including the largest one.

Given Data / Assumptions:
• There are 4 consecutive odd natural numbers.• The average of these 4 numbers is 64.• The numbers are increasing and each differs from the next by 2.• We need to find the largest of these four odd numbers.

Concept / Approach:
For any set of equally spaced numbers (an arithmetic progression), the average is equal to the middle value. For 4 consecutive odd numbers, the average equals the mean of the two central numbers and also equal to the first number plus 3. Alternatively, we can represent the numbers algebraically and use the average formula to solve for the first number, then compute the largest number.

Step-by-Step Solution:
Step 1: Let the four consecutive odd numbers be x, x + 2, x + 4, and x + 6.Step 2: Their average is given as 64.Step 3: Average of these numbers = (x + (x + 2) + (x + 4) + (x + 6)) / 4.Step 4: Simplify the numerator: x + x + 2 + x + 4 + x + 6 = 4x + 12.Step 5: So average = (4x + 12) / 4 = x + 3.Step 6: Set this equal to 64: x + 3 = 64.Step 7: Solve for x: x = 64 - 3 = 61.Step 8: The largest number is x + 6 = 61 + 6 = 67.

Verification / Alternative check:
The four odd numbers are 61, 63, 65, and 67. Their sum is 61 + 63 + 65 + 67 = 256. The average is 256 / 4 = 64, which matches the given value. This confirms that 67 is indeed the largest number.

Why Other Options Are Wrong:
65: This would correspond to a lower average than 64 when taken with three smaller odd numbers.69: This would give an average greater than 64 if used as the largest in the sequence.71: This is even larger and cannot match the average of 64 for four consecutive odd numbers.

Common Pitfalls:
Students sometimes confuse even and odd sequences or assume that the average equals one of the given numbers without checking the arithmetic progression. Others may forget that the step size is 2 for odd numbers, not 1. Using algebra or the property that the average of equally spaced numbers is the central value helps avoid these errors.

Final Answer:
The largest number is 67.