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

Introduction / Context:
This question focuses on consecutive natural numbers and their averages. When numbers are consecutive, they form an arithmetic progression with common difference 1. The average of such a sequence has a simple relationship to the first and last terms. We are given the average and asked to recover the largest of the 8 consecutive numbers.

Given Data / Assumptions:

• There are 8 consecutive natural numbers.
• Their average is 38.5.
• We must find the largest of these 8 numbers.

Concept / Approach:
For an arithmetic progression, the average is equal to the midpoint between the first and last terms. For 8 consecutive numbers, if the first term is x, the numbers are x, x+1, x+2, ..., x+7, and the last term is x+7. The average is then (x + (x + 7)) / 2 = (2x + 7) / 2. We can set this equal to 38.5 and solve for x. Once the first term is found, the last term is x + 7. This is a direct linear equation and does not require listing all numbers explicitly.

Step-by-Step Solution:
Step 1: Let the first of the 8 consecutive numbers be x. Then the numbers are x, x + 1, x + 2, ..., x + 7. The last number is x + 7. Step 2: Use the average formula for an arithmetic progression. Average = (first term + last term) / 2. Given average = 38.5, so (x + (x + 7)) / 2 = 38.5. This simplifies to (2x + 7) / 2 = 38.5. Step 3: Multiply both sides by 2. 2x + 7 = 77. Step 4: Solve for x. 2x = 77 − 7 = 70. x = 70 / 2 = 35. Step 5: Find the largest number. Largest number = x + 7 = 35 + 7 = 42.
Verification / Alternative check:
The 8 numbers are 35, 36, 37, 38, 39, 40, 41, 42. Sum = 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42. Pairs from extremes sum to 77: (35 + 42), (36 + 41), (37 + 40), (38 + 39). Each pair sums to 77 and there are 4 pairs, so total sum = 4 * 77 = 308. Average = 308 / 8 = 38.5, matching the given average.
Why Other Options Are Wrong:
If the largest number were 41, then the first would be 34, and the average would be (34 + 41) / 2 = 37.5, not 38.5. For largest 39 or 40, the averages would be even smaller, while 45 would produce a much higher average than 38.5.
Common Pitfalls:
Some students try to compute the numbers starting from the average by adding and subtracting 1 without using a systematic equation, which can lead to mistakes. Others incorrectly use average = last term − first term, which is not a valid formula.
Final Answer:
The largest of the 8 consecutive natural numbers is 42.