Understanding the Arithmetic Mean of Natural Numbers

The question asks for the arithmetic mean of the first \(n\) natural numbers. Natural numbers are the counting numbers starting from 1. So, the sequence of numbers is 1, 2, 3, 4, ..., \(n\).

What is Arithmetic Mean?

The arithmetic mean, also known as the average, is a measure of central tendency. It is calculated by summing up all the numbers in a set and then dividing the sum by the total count of numbers in that set.

The formula for the arithmetic mean is:

Arithmetic Mean = \(\frac{\text{Sum of all numbers}}{\text{Total count of numbers}}\)

Calculating the Sum of the First \(n\) Natural Numbers

We need to find the sum of the numbers 1, 2, 3, ..., \(n\). This is a sum of an arithmetic progression with the first term \(a_1 = 1\), the last term \(a_n = n\), and the number of terms equal to \(n\). The sum of the first \(n\) natural numbers can be calculated using the sum formula:

\(\text{Sum} = \frac{n \times (\text{First term} + \text{Last term})}{2}\)

\(\text{Sum} = \frac{n \times (1 + n)}{2}\)

So, the sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). This is an important sum formula to remember for calculating averages and other statistical measures involving natural numbers.

Finding the Arithmetic Mean

Now that we have the sum of the numbers and the total count of numbers (which is \(n\)), we can calculate the arithmetic mean using the formula:

Arithmetic Mean = \(\frac{\text{Sum of the first \(n\) natural numbers}}{\text{Total count of numbers}}\)

Arithmetic Mean = \(\frac{\frac{n(n+1)}{2}}{n}\)

To simplify this expression, we can write \(n\) as \(\frac{n}{1}\) and perform the division:

Arithmetic Mean = \(\frac{n(n+1)}{2} \div \frac{n}{1}\)

Arithmetic Mean = \(\frac{n(n+1)}{2} \times \frac{1}{n}\)

We can cancel out the \(n\) from the numerator and the denominator:

Arithmetic Mean = \(\frac{\cancel{n}(n+1)}{2 \times \cancel{n}}\)

Arithmetic Mean = \(\frac{n+1}{2}\)

Thus, the arithmetic mean of the first \(n\) natural numbers is \(\frac{n+1}{2}\).

Summary of the Mean Calculation

To find the arithmetic mean of the first \(n\) natural numbers:

• Identify the numbers: 1, 2, 3, ..., \(n\).
• Find the sum using the sum formula: \(\frac{n(n+1)}{2}\).
• Identify the count: \(n\).
• Divide the sum by the count to get the arithmetic mean: \(\frac{n+1}{2}\).

This method provides a clear way to calculate the average for any sequence of the first \(n\) natural numbers.

The arithmetic mean of 1, 2, 3, ..., \(n\) numbers is \(\frac{n+1}{2}\).