What is the average of the first 19 odd natural numbers?

Introduction / Context:
This question is about averages and the sequence of odd natural numbers. The first few odd numbers are 1, 3, 5, and so on. You are asked to find the average of the first 19 such numbers. This problem highlights an elegant property of consecutive odd numbers and reinforces basic understanding of arithmetic sequences and averages.

Given Data / Assumptions:
- The sequence considered is the sequence of odd natural numbers: 1, 3, 5, 7, ...
- We are interested in the first 19 odd numbers in this sequence.
- We must compute their average value.

Concept / Approach:
The nth odd number can be written as 2n - 1. Thus, the first 19 odd numbers are 1, 3, 5, ..., 2 * 19 - 1 = 37. For any equally spaced sequence that forms an arithmetic progression, the average of all terms is equal to the average of the first and last terms. Since the odd numbers form an arithmetic progression with common difference 2, we can directly use this property without summing all 19 numbers individually.

Step-by-Step Solution:
Step 1: Identify the first odd natural number in the sequence, which is 1.Step 2: The nth odd number is 2n - 1. For n = 19, the 19th odd number is 2 * 19 - 1 = 38 - 1 = 37.Step 3: Therefore, the first 19 odd numbers are 1, 3, 5, ..., 37. This is an arithmetic progression with first term 1 and last term 37.Step 4: The average of an arithmetic progression is equal to (first term + last term) / 2.Step 5: Compute this average: (1 + 37) / 2 = 38 / 2 = 19.Step 6: Hence, the average of the first 19 odd natural numbers is 19.

Verification / Alternative check:
As a quick check, consider the pattern for smaller sets. The average of the first odd number (1) is 1, which equals 1. The average of the first 3 odd numbers 1, 3, and 5 is (1 + 3 + 5) / 3 = 9 / 3 = 3, which equals the middle term and also equals the total count of numbers. For 5 odd numbers, 1 + 3 + 5 + 7 + 9 = 25, average = 25 / 5 = 5. The pattern suggests that the average of the first n odd numbers is always n. So for n = 19, the average is 19, which matches our earlier calculation.

Why Other Options Are Wrong:
Values such as 9.5 or 15.5 or 38 result from misinterpreting the range or misapplying the average formula. For example, 38 is the sum of the first and last terms, not the average, and 9.5 is half of 19 but has no direct meaning in this context. The number 20 would be the average of 1 and 39, which is not the last number in the sequence here, so it is also incorrect.

Common Pitfalls:
Some learners think they need to sum all 19 odd numbers individually, which is time consuming and prone to arithmetic errors. Others mistakenly take the last term 37 or the count 19 as the sum or confuse them. Remembering the property that the average of consecutive terms in an arithmetic progression equals the average of the first and last terms simplifies the calculation greatly.

Final Answer:
The average of the first 19 odd natural numbers is 19.