Question

Find the average of first 50 natural numbers:

• A. 25.5
• B. 25
• C. 35.5
• D. 35

Hint:

For any evenly spaced arithmetic progression, the average is simply the average of the first and the last term: \[ \text{Average} = \frac{\text{First Term} + \text{Last Term}}{2} \] Here: $\frac{1 + 50}{2} = 25.5$ directly.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The average of a set of numbers is calculated by dividing the sum of the elements by the total count of elements.
Natural numbers starting from 1 up to $n$ form a consecutive arithmetic sequence.
Key Formula or Approach:
The formula for the sum ($S_n$) of the first $n$ natural numbers is: \[ S_n = \frac{n(n + 1)}{2} \] The average ($A_n$) of the first $n$ natural numbers is: \[ A_n = \frac{S_n}{n} = \frac{n(n + 1)}{2n} = \frac{n + 1}{2} \]

Step 2: Detailed Explanation:
We are asked to find the average of the first 50 natural numbers.
Here, the number of terms $n = 50$.
Substitute $n = 50$ directly into our average formula: \[ A_{50} = \frac{50 + 1}{2} \] \[ A_{50} = \frac{51}{2} \] \[ A_{50} = 25.5 \] Let us double-check using the sum of the series: \[ \text{Sum} = 1 + 2 + 3 + \dots + 50 \] \[ \text{Sum} = \frac{50 \times 51}{2} = 25 \times 51 = 1275 \] \[ \text{Average} = \frac{\text{Sum}}{50} = \frac{1275}{50} = 25.5 \] Both methods yield the same result.

Step 3: Final Answer:
The average of the first 50 natural numbers is 25.5.