What is the average of the first 39 even natural numbers?

Introduction / Context:
This question involves the average of the first several even numbers. It tests understanding of arithmetic progressions and the property that evenly spaced numbers have an average equal to the midpoint of the first and last terms.

Given Data / Assumptions:
- We are considering the first 39 even natural numbers. - The first even number is 2. - The numbers increase by 2 each time (common difference = 2). - We need the arithmetic mean of these 39 numbers.

Concept / Approach:
The first n even numbers form an arithmetic progression: 2, 4, 6, ... . The nth even number equals 2n. For an arithmetic progression with first term a and last term l, the average is:
average = (a + l) / 2. Here a = 2 and the 39th even number l = 2 * 39. We use this formula directly to get the mean, which is much faster than summing each term.

Step-by-Step Solution:
Step 1: Identify the first term: a = 2. Step 2: Identify the 39th even number. Since the nth even number is 2n, the 39th even number is 2 * 39 = 78. Step 3: Use the formula for the average of an arithmetic progression: average = (a + l) / 2. Step 4: Substitute a = 2 and l = 78: average = (2 + 78) / 2. Step 5: Compute sum: 2 + 78 = 80. Step 6: Divide by 2: 80 / 2 = 40. Step 7: Therefore, the average of the first 39 even numbers is 40.

Verification / Alternative check:
Alternatively, we can use the sum formula. Sum of n terms of an arithmetic progression is n * (a + l) / 2. Here n = 39, a = 2, l = 78, so total sum = 39 * 80 / 2 = 39 * 40 = 1560. Average = total sum / n = 1560 / 39. Dividing 1560 by 39 gives 40 exactly, confirming that the average is correct.

Why Other Options Are Wrong:
Option A (39): Close, but it would be the mean if numbers were symmetric around 39, which is not the case here. Option C (20): Much too small, roughly the mean of the first 20 even numbers rather than 39. Option D (68): Too large and not equal to the midpoint of 2 and 78. Option E (38): Also not equal to the arithmetic mean of 2 and 78.

Common Pitfalls:
Many learners mistakenly assume that the average of the first n even numbers is n, which is not correct. Others misidentify the last term and might think it is 2 * 38 instead of 2 * 39. The safest method is always to use the formula for the nth term and then apply the average formula for arithmetic progressions.

Final Answer:
The average of the first 39 even natural numbers is 40.