The average of 8 consecutive natural numbers is 38.5. What is the largest of these 8 numbers?

Introduction / Context:
This problem uses properties of consecutive natural numbers and their average. When numbers are consecutive, the average lies exactly in the middle of the first and last terms, which helps us reconstruct the sequence from the average.

Given Data / Assumptions:
- There are 8 consecutive natural numbers. - Their average is 38.5. - We must determine the largest of the 8 numbers.

Concept / Approach:
Let the first number be x, then the sequence is x, x + 1, x + 2, ..., x + 7. For any arithmetic progression, the average equals the mean of first and last terms. Thus average = (x + x + 7) / 2 = x + 3.5. Setting this equal to 38.5 allows us to solve for x and then find x + 7, the largest number in the list.

Step-by-Step Solution:
Step 1: Represent the 8 numbers as x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7. Step 2: The first term is x and the last term is x + 7. Step 3: Average of an arithmetic progression = (first + last) / 2. Step 4: So average = (x + (x + 7)) / 2 = (2x + 7) / 2 = x + 3.5. Step 5: Given average is 38.5, so x + 3.5 = 38.5. Step 6: Solve: x = 38.5 - 3.5 = 35. Step 7: Largest number = x + 7 = 35 + 7 = 42.

Verification / Alternative check:
List the numbers explicitly: 35, 36, 37, 38, 39, 40, 41 and 42. Their sum is 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 308. Average = 308 / 8 = 38.5, which matches the given value and confirms that 42 is indeed the largest number.

Why Other Options Are Wrong:
- 39, 40 and 41 are smaller than the correct largest term for this sequence. - 45 is too large; including such a number would shift the average above 38.5.

Common Pitfalls:
Learners sometimes mistakenly treat 38.5 as one of the terms or misplace the largest term by one or two positions. Remember that for an even number of consecutive terms, the average lies between the two middle terms, not necessarily equal to one of them. Using the first and last term relation is more reliable.

Final Answer:
The largest of the 8 consecutive natural numbers is 42.