Average Calculation for First 169 Even Numbers

This solution explains how to find the average of the first 169 positive even numbers using clear methods and mathematical formulas.

Understanding Even Numbers

Even numbers are integers that are divisible by 2 without leaving a remainder. The sequence of positive even numbers starts like this:

• 2
• 4
• 6
• 8
• 10
• ...

The general formula for the $n$-th positive even number is $2n$. For instance, the 10th even number is $2 \times 10 = 20$.

Identifying the Sequence

To find the average of the first 169 even numbers, we need to consider the sequence from the 1st even number up to the 169th even number.

The 1st even number is $2 \times 1 = 2$.

The 169th even number is calculated as $2 \times 169 = 338$.

So, the sequence we are averaging is: 2, 4, 6, ..., 338.

Calculating the Average

The average is the sum of all numbers divided by the count of the numbers. We can calculate the average of the first 169 even numbers using a couple of methods:

Method 1: Using the Direct Formula

There is a straightforward formula for the average of the first $n$ positive even numbers: Average = $n + 1$.

In this case, we are looking for the average of the first 169 even numbers, so $n = 169$.

Applying the formula:

Average = $169 + 1$

Average = $170$

This method is quick and efficient for finding the average of consecutive even numbers.

Method 2: Using Arithmetic Progression (AP) Properties

The sequence of the first 169 even numbers (2, 4, 6, ..., 338) forms an arithmetic progression.

• The first term ($a$) is 2.
• The last term ($l$) is 338.
• The total number of terms ($n$) is 169.

The average of an arithmetic progression is found by taking the average of the first and the last term:

Average = $\frac{\text{First Term} + \text{Last Term}}{2} = \frac{a + l}{2}$

Substituting the values:

Average = $\frac{2 + 338}{2}$

Average = $\frac{340}{2}$

Average = $170$

Both methods confirm that the average of the first 169 even numbers is 170.