What is the average of the first 39 even natural numbers?

Introduction / Context:
This problem asks for the average of the first 39 even natural numbers. Even numbers follow a very regular pattern, so this is a straightforward application of the concept of averages in an arithmetic progression. Such questions are common in competitive exams to test quick calculation skills and understanding of sequences.

Given Data / Assumptions:
- We consider the first 39 even natural numbers.
- The sequence of even natural numbers is 2, 4, 6, 8, and so on, with common difference 2.
- We need to find the arithmetic mean of these 39 even numbers.

Concept / Approach:
The first 39 even natural numbers form an arithmetic progression where the first term is 2 and the common difference is 2. The nth term of this progression is 2n. For n = 39, the last term is 2 * 39 = 78. For any arithmetic progression, the average equals the mean of the first and last terms, because the terms can be paired symmetrically around the center.

Step-by-Step Solution:
Step 1: Identify the first even number in the list: 2.Step 2: Identify the last even number in the list, which is the 39th even number.Step 3: The nth even number is 2n, so the 39th even number is 2 * 39 = 78.Step 4: Use the arithmetic mean formula for an arithmetic progression: average = (first term + last term) / 2.Step 5: Substitute the values: average = (2 + 78) / 2.Step 6: Compute the sum: 2 + 78 = 80.Step 7: Divide by 2: 80 / 2 = 40.Step 8: Therefore, the average of the first 39 even natural numbers is 40.

Verification / Alternative check:
You can pair the numbers from opposite ends: (2, 78), (4, 76), (6, 74), and so on. Each pair sums to 80, so the average of each pair is 80 / 2 = 40. Since all pairs share the same average and the list is made entirely from such pairs, the overall average of the full set must also be 40. This further supports the calculated answer.

Why Other Options Are Wrong:
Values such as 39 or 20 might be chosen if someone splits or counts the numbers incorrectly, while 68 or 78 are closer to the higher end of the range and cannot possibly represent the average of numbers that start from 2. Only 40 lies exactly in the middle of the first and last term, which is the correct mean for this arithmetic progression.

Common Pitfalls:
Some students mistakenly think that the average of the first 39 even numbers should be equal to the 20th term or the common difference. Others forget to identify the last term correctly as 78. Always remember the simple rule: for evenly spaced numbers, the average is the midpoint between the smallest and largest term.

Final Answer:
The average of the first 39 even natural numbers is 40.