Question

The arithmetic mean of \(1,2,3,\ldots,n\) is:

• A. \(\dfrac{n+1}{2}\)
• B. \(\dfrac{n-1}{2}\)
• C. \(\dfrac{n}{2}\)
• D. \(\dfrac{2n+1}{2}\)

Hint:

For any arithmetic progression, \[ \text{Mean} = \frac{\text{First Term}+\text{Last Term}}{2}. \] Thus, \[ \frac{1+n}{2} = \frac{n+1}{2}. \]

Answer 1

The Correct Option is A

Concept: The arithmetic mean is \[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}. \] Also, \[ 1+2+3+\cdots+n = \frac{n(n+1)}{2}. \]

Step 1: Find the sum of the first \(n\) natural numbers. \[ S_n = 1+2+3+\cdots+n = \frac{n(n+1)}{2}. \]

Step 2: Apply the mean formula. Since there are \(n\) numbers, \[ \text{Mean} = \frac{S_n}{n} = \frac{\frac{n(n+1)}{2}}{n}. \] Cancelling \(n\), \[ \text{Mean} = \frac{n+1}{2}. \] Conclusion: \[ \boxed{\frac{n+1}{2}} \] Hence, the correct answer is Option (A).