What is the average (arithmetic mean) of the first 11 multiples of 11?

Introduction / Context:
This question involves finding the average of the first 11 multiples of 11. These numbers form an arithmetic progression, and there is a simple way to find their average without listing every term. This tests understanding of averages and the structure of arithmetic sequences.

Given Data / Assumptions:
The first 11 multiples of 11 are 11, 22, 33, ..., 121.We must compute their average (sum divided by 11).This is an arithmetic progression with common difference 11.

Concept / Approach:
In an arithmetic progression, the average of the first n terms is equal to the average of the first and last term, that is, (first term + last term) / 2. Since these are consecutive multiples of 11, we can identify the first multiple (11) and the 11th multiple (11 × 11 = 121) and use this property directly.

Step-by-Step Solution:
Step 1: List enough terms to see the pattern: 11, 22, 33, ..., 121.Step 2: Recognize that this is an arithmetic progression with first term a = 11 and common difference d = 11.Step 3: The 11th term is 11 × 11 = 121.Step 4: Use the fact that the average of an arithmetic series is (first term + last term) / 2.Step 5: Compute average = (11 + 121) / 2.Step 6: Add the numerator: 11 + 121 = 132.Step 7: Divide by 2: 132 / 2 = 66.

Verification / Alternative check:
We can also compute the sum explicitly. The sum S of n terms of an A.P. is S = n * (first term + last term) / 2. Here n = 11, a = 11 and last term l = 121. So S = 11 * (11 + 121) / 2 = 11 * 132 / 2 = 11 * 66 = 726. The average is S / 11 = 726 / 11 = 66, confirming the result.

Why Other Options Are Wrong:
The value 22 is the second multiple of 11, not the average. The values 44 and 55 are mid range but not the exact center of the sequence. Because the sequence runs from 11 to 121, the true middle value, and hence the average, must be halfway between 11 and 121, which is 66.

Common Pitfalls:
Some learners mistakenly think the average is simply the 6th multiple of 11 (which is 66 here by coincidence) without realizing why this works. While that gives the correct numerical answer in this case, the more general and reliable method is using (first term + last term) / 2 or the full sum formula.

Final Answer:
The average of the first 11 multiples of 11 is 66.