What is the average of all natural numbers from 21 to 39 (both inclusive)?

Introduction / Context:
This is an average question involving consecutive natural numbers in a closed interval. It checks whether you know that for evenly spaced numbers, the average can be found quickly without summing each term by using the simple midpoint formula of the first and last term.

Given Data / Assumptions:
- We are dealing with natural numbers from 21 to 39 inclusive. - The numbers form an arithmetic progression with common difference 1. - We need to find their arithmetic mean (average).

Concept / Approach:
For any arithmetic progression where terms are equally spaced, the average of all the terms is simply equal to the mean of the first and last term. That is,
average = (first term + last term) / 2. This formula saves time, as it avoids computing the sum and dividing by the count explicitly. Since natural numbers 21 to 39 are consecutive, this concept applies directly.

Step-by-Step Solution:
Step 1: Identify the first term of the sequence: 21. Step 2: Identify the last term of the sequence: 39. Step 3: Use the average formula for an arithmetic progression: average = (first term + last term) / 2. Step 4: Substitute values: average = (21 + 39) / 2. Step 5: Compute the sum of first and last: 21 + 39 = 60. Step 6: Divide by 2: 60 / 2 = 30. Step 7: Therefore, the required average is 30.

Verification / Alternative check:
As an alternative, we could compute the number of terms and the total sum. Number of terms n = last - first + 1 = 39 - 21 + 1 = 19. The sum of an arithmetic series is n * (first + last) / 2, which gives 19 * 60 / 2 = 19 * 30 = 570. The average is then total sum / n = 570 / 19 = 30. This confirms the earlier result and provides a consistency check.

Why Other Options Are Wrong:
Option B (31): Slightly higher than the true mid value of 30, so it cannot be the average of symmetric numbers around 30. Option C (29): Slightly lower and would correspond to an interval shifted down, not 21 to 39. Option D (28): Not the midpoint of 21 and 39 and does not match the arithmetic progression formula. Option E (32): Also not equal to (21 + 39) / 2, so it is not correct.

Common Pitfalls:
Some learners try to list and sum all terms, which is time consuming and error prone. Others might miscount the number of terms or forget that the average of equally spaced numbers is the midpoint of the first and last term. Remember that using arithmetic progression properties simplifies this type of problem significantly in competitive exams.

Final Answer:
The average of all natural numbers from 21 to 39 inclusive is 30.