What is the average of the first 93 natural numbers (that is, the numbers from 1 to 93 inclusive)?

Introduction / Context:
This question checks understanding of averages and arithmetic progressions. The first 93 natural numbers form a simple series starting at 1 and increasing by 1 each time. Instead of adding all 93 numbers one by one, we can use the properties of this sequence to find the average quickly and accurately.

Given Data / Assumptions:

• We are dealing with the first 93 natural numbers.
• The numbers are 1, 2, 3, ..., 93.
• The sequence is an arithmetic progression with common difference 1.
• We are asked to find the average value of these 93 numbers.

Concept / Approach:
For an arithmetic progression with first term a and last term l, and with an odd or even number of terms, the average of all terms is equal to (a + l) / 2. This is because the sum of the terms is given by n * (a + l) / 2 and the average is that sum divided by n, which simplifies to (a + l) / 2. Here the first term is 1 and the last term is 93, so the average is simply the midpoint between these two numbers.

Step-by-Step Solution:
Step 1: Identify the first and last terms of the sequence. First term a = 1. Last term l = 93. Number of terms n = 93. Step 2: Use the average formula for an arithmetic progression. Average = (first term + last term) / 2. Average = (1 + 93) / 2. Step 3: Compute the sum in the numerator. 1 + 93 = 94. Step 4: Divide by 2. Average = 94 / 2 = 47.
Verification / Alternative check:
If we wanted, we could confirm by using the sum formula. Sum of first 93 natural numbers = n * (a + l) / 2 = 93 * 94 / 2. Compute 93 * 94 = 8742 and then divide by 2 to get 4371. Average = total sum / number of terms = 4371 / 93 = 47, which confirms the earlier result.
Why Other Options Are Wrong:
Values like 45, 46, 48 or 49 correspond to midpoints between different first and last numbers and do not match the exact midpoint of 1 and 93. Any other value would violate the property that, in a symmetric sequence from 1 to 93, the average must lie exactly in the middle.
Common Pitfalls:
Students sometimes think they must add all 93 numbers explicitly, which is unnecessary and error prone. Another error is to divide the last term 93 by 2 instead of taking the average of first and last terms, which gives 46.5 and is not correct for this question.
Final Answer:
The average of the first 93 natural numbers is 47.