Find the average of the first 50 natural numbers (1, 2, 3, ..., 50).

Introduction / Context:
This question asks for the average of the first 50 natural numbers. It is a standard problem involving arithmetic progressions. The first n natural numbers form a simple arithmetic sequence with a well known sum formula, and their average can be obtained directly using symmetry or the sum formula divided by n.

Given Data / Assumptions:

• The numbers are 1, 2, 3, ..., 50.

• These numbers form an arithmetic progression with first term 1 and last term 50.

• We are using the usual definition of natural numbers starting from 1.

Concept / Approach:
For an arithmetic progression with first term a, last term l and n terms, the average is simply (a + l) / 2. This is because the terms are symmetrically placed around the mid value. Therefore, for the first 50 natural numbers, the average is (1 + 50) / 2. Alternatively, we can use the sum formula n * (a + l) / 2 and then divide by n, which yields the same result for the mean.

Step-by-Step Solution:
First term a = 1. Last term l = 50. Number of terms n = 50. Average of an arithmetic progression = (first term + last term) / 2. So, average = (1 + 50) / 2. Average = 51 / 2 = 25.5.

Verification / Alternative check:
We can verify by using the sum formula. Sum of first n natural numbers = n * (n + 1) / 2. For n = 50, sum = 50 * 51 / 2 = 25 * 51 = 1275. Average = sum / n = 1275 / 50 = 25.5. This matches the simple average formula result, confirming the correctness.

Why Other Options Are Wrong:
An average of 25 would correspond to numbers symmetric around 25, for example from 1 to 49, not 1 to 50. Averages of 26 or 26.5 would require a higher centre than 25.5 and do not match the actual spread of the numbers from 1 to 50. Only 25.5 is exactly mid way between 1 and 50 and is backed by the arithmetic progression formulas.

Common Pitfalls:
Students may mistakenly add 50 and 1 and divide by something other than 2, or confuse the formula with that for the sum of the first n numbers. Another typical confusion is thinking that the average equals n / 2 (which here would be 25), ignoring the shift due to the sequence starting at 1. Always remember that for 1 to n, the average is (1 + n) / 2, not simply n / 2.

Final Answer:
The average of the first 50 natural numbers is 25.5.