Finding the Mean of the First 60 Natural Numbers

The question asks us to find the mean of the first 60 natural numbers. Natural numbers are positive integers starting from 1.

The first 60 natural numbers are: 1, 2, 3, ..., 60.

To find the mean of a set of numbers, we use the formula:

$$ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}} $$

In this case, the total count of numbers is 60.

Calculating the Sum of the First 60 Natural Numbers

The first 60 natural numbers form an arithmetic progression with the first term \(a_1 = 1\), the last term \(a_{60} = 60\), and the number of terms \(n = 60\).

The sum of the first \(n\) natural numbers can be found using the formula:

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

Using this formula for \(n=60\):

$$ \text{Sum of first 60 natural numbers} = \frac{60(60+1)}{2} $$

$$ \text{Sum} = \frac{60 \times 61}{2} $$

$$ \text{Sum} = 30 \times 61 $$

$$ \text{Sum} = 1830 $$

So, the sum of the first 60 natural numbers is 1830.

Calculating the Mean

Now we can calculate the mean using the sum (1830) and the total count of numbers (60):

$$ \text{Mean} = \frac{1830}{60} $$

We can simplify the fraction:

$$ \text{Mean} = \frac{183}{6} $$

Now, performing the division:

$$ \text{Mean} = 30.5 $$

Thus, the mean of the first 60 natural numbers is 30.5.

Step-by-Step Summary to find Mean of Natural Numbers

1. Identify the number of natural numbers, n (here, n=60).
2. Calculate the sum of the first n natural numbers using the formula \(\text{Sum} = \frac{n(n+1)}{2}\).
3. Calculate the mean by dividing the sum by the number of terms: \(\text{Mean} = \frac{\text{Sum}}{n}\).

Applying these steps for the first 60 natural numbers:

1. n = 60.
2. Sum = \(\frac{60(60+1)}{2} = \frac{60 \times 61}{2} = 30 \times 61 = 1830\).
3. Mean = \(\frac{1830}{60} = 30.5\).

The result, 30.5, matches one of the given options.

Revision Table: Key Concepts for Mean Calculation

Additional Information: Mean and Arithmetic Progressions

The set of first natural numbers is an example of an arithmetic progression (AP). An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. In this case, the constant difference is 1.

For any arithmetic progression, the mean is equal to the average of the first and last term.

Let's check this for the first 60 natural numbers:

• First term (\(a_1\)) = 1
• Last term (\(a_{60}\)) = 60
• Average of first and last term = \(\frac{a_1 + a_{60}}{2}\)
• Average = \(\frac{1 + 60}{2} = \frac{61}{2} = 30.5\)

This confirms the result obtained using the sum formula. This shortcut works specifically for arithmetic progressions.

Understanding the properties of arithmetic progressions can simplify finding the mean for sequences like natural numbers, even numbers, odd numbers, etc., when they form an AP.