What is the mean of first hundred natural numbers?
50.5
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.
| Step | Description | Value |
|---|---|---|
| 1 | Identify 'n' (number of natural numbers) | 100 |
| 2 | Calculate the sum using the formula \( \frac{n(n+1)}{2} \) | \( \frac{100(101)}{2} = 5050 \) |
| 3 | Identify the total count of numbers | 100 |
| 4 | Calculate the mean using \( \frac{\text{Sum}}{\text{Count}} \) | \( \frac{5050}{100} = 50.5 \) |
The calculated mean is 50.5.
Revision Table: Mean of First Hundred Natural Numbers
| Concept | Definition/Formula |
|---|---|
| Natural Numbers | Counting numbers starting from 1 (1, 2, 3, ...) |
| Mean (Arithmetic Average) | \( \frac{\text{Sum of observations}}{\text{Number of observations}} \) |
| Sum of first 'n' Natural Numbers | \( \frac{n(n+1)}{2} \) |
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.
Important Questions from Average
The average of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
