Question

The average of first 100 natural number is : (a) 100 (b) 50 (c) \(50.50\) (d) 55

Short answer

Answer: The average of the first 100 natural numbers is 50.50.

Step-by-step solution

Identify the formula for the sum of the first n natural numbers

The formula for the sum of the first n natural numbers is: $$ S = \frac{n(n+1)}{2} $$

Calculate the sum of the first 100 natural numbers

Using the formula above, let's calculate the sum of the first 100 natural numbers. Plug in n = 100 into the formula: $$ S = \frac{100(100 + 1)}{2} $$ $$ S = \frac{100 \cdot 101}{2} $$ $$ S = 50 \cdot 101 $$ $$ S = 5050 $$

Calculate the average of the first 100 natural numbers

Now that we've calculated the sum of the first 100 natural numbers, we need to divide this sum by the total number of numbers (100) to get the average: $$ \text{Average} = \frac{\text{Sum}}{\text{Number of elements}} $$ $$ \text{Average} = \frac{5050}{100} = 50.50 $$

Choose the correct answer

The average of the first 100 natural numbers is \(50.50\). So, the correct answer is (c) \(50.50\).

Key concepts

Sum of Natural Numbers Formula

The sum of natural numbers is an important topic in mathematics and serves as a foundation for various calculations. To find the sum of the first \( n \) natural numbers, we use a simple yet powerful formula:\[ S = \frac{n(n+1)}{2} \]This formula might seem mysterious at first, but it's quite intuitive. Imagine a long list of numbers from 1 to \( n \). The purpose of the formula is to provide a quick method to compute the sum of these numbers without having to add them one by one. Consider a case where \( n = 100 \). To find the sum of all numbers from 1 to 100, substitute \( n \) into the formula:\[ S = \frac{100(100 + 1)}{2} = \frac{100 \times 101}{2} = 5050 \]By breaking this down, adding 1 through 100 manually would be tedious, but the formula simplifies the work beautifully.

Arithmetic Mean

Arithmetic mean, often referred to as average, is a fundamental concept in statistics and everyday math problems. It is essentially the central value of a set of numbers. To calculate the arithmetic mean from a given set of numbers, follow this simple approach:1. Compute the sum of the numbers in the set.2. Divide the sum by the total number of values in the set.For the first 100 natural numbers, we already calculated the sum as 5050. Since there are 100 numbers in this set, calculating the average involves dividing this total sum by 100:\[ \text{Average} = \frac{5050}{100} = 50.50 \]This tells us that the arithmetic mean of the first 100 natural numbers is 50.50. It is important to note that this concept helps to comprehend the "typical" value in a set of numbers, providing a simple measure of central tendency.

Numerical Calculation

Numerical calculation is the aspect of arithmetic that deals with computing or solving problems using numbers. It forms the core of solving textbook exercises like finding the average of numbers. Calculations often involve:

• Breaking down the required steps logically.
• Performing operations as per mathematical rules.
• Verifying results for accuracy.

Let's reflect on the task of calculating the average of the first 100 natural numbers. The numerical calculations involved were straightforward due to the formula and steps outlined. We used the formula to compute the sum (5050) and divided by the count of the numbers (100) to achieve the final average of 50.50. Understanding numerical calculation helps in efficiently processing numbers, ensuring results are correct, and simplifying complex mathematical exercises.