What is the average of the first 19 odd natural numbers (1, 3, 5, …)?

Introduction / Context:
This question tests your understanding of sequences, in particular the sequence of odd natural numbers, and the concept of averages in arithmetic progressions. The first 19 odd natural numbers form a simple arithmetic progression, and their average can be found without listing all numbers individually.

Given Data / Assumptions:

• The first odd natural number is 1.
• The sequence of odd numbers is: 1, 3, 5, 7, …
• We are considering the first 19 terms of this sequence.
• We need the average of these 19 numbers.

Concept / Approach:
The first 19 odd numbers form an arithmetic progression (AP) with: first term a₁ = 1, common difference d = 2. For an AP with an odd number of terms, the average is equal to the middle term. Also, the nth odd number is given by 2n − 1. For n = 19, the last term is 2 * 19 − 1. Once we know the first and last terms, we can also use the formula for the average of an AP: average = (first term + last term) / 2.

Step-by-Step Solution:
Step 1: Identify the first and last of the first 19 odd numbers. First term a₁ = 1. The nth odd number = 2n − 1, so the 19th odd number = 2 * 19 − 1 = 38 − 1 = 37. Step 2: Use the average formula for an arithmetic progression. Average = (first term + last term) / 2. Average = (1 + 37) / 2 = 38 / 2 = 19. Step 3: Alternatively, use the middle-term idea. Since there are 19 terms (an odd number), the average is equal to the 10th term (the middle term). The 10th odd number is 2 * 10 − 1 = 19, confirming the same result.
Verification / Alternative check:
We can quickly list a smaller case to see the pattern: For the first 3 odd numbers 1, 3, 5, the average is (1 + 3 + 5)/3 = 9/3 = 3, which equals the middle term. Similarly, with 5 odd numbers 1, 3, 5, 7, 9, the average is 5, the middle term. The same principle scales to 19 terms.
Why Other Options Are Wrong:
9.5 and 15.5 are too small and would be the average of fewer or different odd numbers, not the first 19. 38 is the last term, not the average. 20 is slightly larger than 19 and would require a different first or last term.
Common Pitfalls:
Some students mistakenly average the index 19 with 1 instead of the values of the numbers. Others may divide the last term 37 by 2 and misinterpret that as the average of the series, rather than using the correct formula (1 + 37)/2.
Final Answer:
The average of the first 19 odd numbers is 19.