Average of All Positive Even Integers Less Than or Equal to 40

Understanding the Given Average Problem

This question asks us to calculate the average of all positive even integers that are less than or equal to 40. To solve the problem correctly, we first need to identify the required numbers and then apply the appropriate formula for calculating their average.

A positive even integer is a positive whole number that is exactly divisible by 2. Therefore, the positive even integers less than or equal to 40 are:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40

These numbers form a regular sequence in which every term is obtained by adding 2 to the previous term. Such a sequence is known as an arithmetic progression or arithmetic sequence.

What Is the Average of a Set of Numbers?

The average, also called the arithmetic mean, represents the central value of a group of numbers. In general, the average is calculated by adding all the observations and dividing the resulting sum by the total number of observations.

The basic formula is:

Average = Sum of all observations / Number of observations

Although we can add all twenty even integers individually and divide their sum by 20, there is a much simpler approach because the given numbers form an arithmetic progression.

Step-by-Step Solution

Step 1: Identify the First and Last Even Integers

The smallest positive even integer is 2. Since the question includes all positive even integers less than or equal to 40, the largest number in the sequence is 40.

Therefore:

First term = 2

and

Last term = 40

Step 2: Use the Average Formula for an Arithmetic Progression

For any arithmetic progression, the average of all its terms is equal to the average of the first term and the last term. This works because the terms are equally spaced.

The formula is:

Average = (First term + Last term) / 2

Substituting the first term as 2 and the last term as 40:

Average = (2 + 40) / 2

Therefore:

Average = 42 / 2

Hence:

Average = 21

Why the Arithmetic Progression Formula Works

The positive even integers from 2 to 40 are equally spaced, with a common difference of 2. Because of this symmetry, numbers from opposite ends of the sequence can be paired together.

For example:

2 + 40 = 42

4 + 38 = 42

6 + 36 = 42

8 + 34 = 42

Every pair formed by taking one number from the beginning and one number from the end has the same sum of 42. Therefore, the average value of each pair is:

42 / 2 = 21

This symmetry shows directly why the average of the entire sequence is 21.

Verification Using the Sum of an Arithmetic Progression

The answer can also be verified by first calculating the total number of terms and then finding their sum. The sequence of positive even integers up to 40 can be written as:

2, 4, 6, …, 40

Since the nth positive even integer is 2n, we can determine the number of terms by solving:

2n = 40

Therefore:

n = 20

Thus, there are 20 positive even integers less than or equal to 40.

The sum of an arithmetic progression is given by:

Sum = n(First term + Last term) / 2

Substituting n = 20, the first term as 2, and the last term as 40:

Sum = 20(2 + 40) / 2

Therefore:

Sum = 20 × 42 / 2

Sum = 420

Now, using the basic average formula:

Average = Sum of all observations / Number of observations

we get:

Average = 420 / 20

Therefore:

Average = 21

Understanding the Central Value of the Sequence

The answer 21 may initially appear unusual because 21 itself is not an even number and does not appear in the given sequence. However, an average does not need to be one of the original observations.

The value 21 lies exactly midway between the smallest even integer 2 and the largest even integer 40. Since the numbers are distributed uniformly and symmetrically around this midpoint, 21 is the arithmetic mean of the complete sequence.

Final Answer

The average of all positive even integers less than or equal to 40 is 21.