Finding the Mean of the First Hundred Natural Numbers

The question asks for the mean of the first hundred natural numbers. Let's break down how to solve this problem.

Understanding Natural Numbers

Natural numbers are the counting numbers starting from 1. The first hundred natural numbers are 1, 2, 3, ..., all the way up to 100.

So, the list of numbers is: 1, 2, 3, 4, ..., 99, 100.

What is the Mean?

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

The formula for the mean is:

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

Calculating the Sum of the First Hundred Natural Numbers

To find the sum of the first 'n' natural numbers, there's a handy formula:

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

In this case, we are dealing with the first hundred natural numbers, so \(n = 100\).

Let's plug \(n=100\) into the formula:

\( \text{Sum} = \frac{100(100+1)}{2} \)

\( \text{Sum} = \frac{100(101)}{2} \)

\( \text{Sum} = \frac{10100}{2} \)

\( \text{Sum} = 5050 \)

So, the sum of the first hundred natural numbers is 5050.

Determining the Total Count of Numbers

We are considering the first hundred natural numbers. This means there are exactly 100 numbers in our set.

\( \text{Total count of numbers} = 100 \)

Calculating the Mean

Now we have the sum (5050) and the count (100). We can use the mean formula:

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

\( \text{Mean} = \frac{5050}{100} \)

\( \text{Mean} = 50.5 \)

Thus, the mean of the first hundred natural numbers is 50.5.

The calculated mean is 50.5.

Revision Table: Mean of First Hundred Natural Numbers

Additional Information: Properties of Natural Numbers and Mean

Natural numbers form the basis of many mathematical concepts. The set of natural numbers is often denoted by \( \mathbb{N} = \{1, 2, 3, \dots\} \).

The mean is a measure of central tendency, giving us a typical value for a dataset. Other measures of central tendency include the median (the middle value) and the mode (the most frequent value).

The formula for the sum of the first 'n' natural numbers was discovered by the mathematician Carl Friedrich Gauss. This formula is very useful for quickly summing arithmetic progressions where the first term is 1 and the common difference is 1.

In this case, the numbers form an arithmetic progression with the first term \(a_1 = 1\) and the last term \(a_{100} = 100\). The mean of an arithmetic progression is simply the average of the first and last terms:

\( \text{Mean} = \frac{a_1 + a_n}{2} = \frac{1 + 100}{2} = \frac{101}{2} = 50.5 \)

This confirms our earlier calculation using the sum formula.