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

Introduction / Context:
This question tests your knowledge of a well known property of the sequence of odd natural numbers. Knowing the pattern of odd numbers helps you quickly compute their averages without listing all of them.

Given Data / Assumptions:
- We consider the first 19 odd natural numbers.
- These are 1, 3, 5, ..., up to the 19th odd number.
- We need to find their average.

Concept / Approach:
The nth odd natural number is given by 2n - 1. So the first 19 odd numbers are 1, 3, 5, ..., 37. A key property is that the average of the first n odd numbers is equal to n itself. This follows from the total sum formula for odd numbers and the definition of average.

Step-by-Step Solution:
Step 1: The nth odd number is 2n - 1, so the 19th odd number is 2 * 19 - 1 = 37. Step 2: The sum of the first n odd numbers is n^2. For n = 19, the sum is 19^2. Step 3: Compute 19^2 = 361. Step 4: There are 19 terms, so the average is sum divided by number of terms = 361 / 19. Step 5: Compute 361 / 19 = 19.

Verification / Alternative check:
You can quickly verify the pattern for smaller values. For n = 3, the first three odd numbers are 1, 3 and 5. Their sum is 9, and average is 9 / 3 = 3, which equals n. The same rule applies for n = 19, so the average must be 19.

Why Other Options Are Wrong:
- Option 9.5: This is less than the middle term of the sequence and cannot be the average of symmetric odd numbers.

- Option 15.5: This is also too low relative to the spread from 1 to 37.

- Option 38: This is larger than every term except the last one and cannot be an average of the whole set.

Common Pitfalls:
Some students try to list and add all 19 numbers, increasing the chance of calculation error. Others misunderstand the pattern and think the average is the middle term in every case, without verifying. In fact, for an arithmetic progression, the average equals the middle term, and the odd numbers form an arithmetic progression. Recognising this gives a quick route to the answer.

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