Question

The mean of consecutive positive integers from 2 to \( n \) is:

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

Hint:

The mean of consecutive integers can be calculated by using the sum formula for an arithmetic sequence.

Answer 1

The Correct Option is A

The mean of the consecutive integers from 2 to \( n \) is calculated by finding the sum of the integers from 2 to \( n \) and dividing by the number of terms.
Step 1: Calculate the sum of the integers from 2 to \( n \).
The sum of consecutive integers from 2 to \( n \) is: \[ \text{Sum} = \frac{n(n+1)}{2} - 1 \quad (\text{since the sum from 1 to } n \text{ is } \frac{n(n+1)}{2} \text{ and we subtract the first term 1}). \] Step 2: Divide by the number of terms.
The number of terms is \( n - 2 + 1 = n - 1 \).
Thus, the mean is: \[ \text{Mean} = \frac{\frac{n(n+1)}{2} - 1}{n - 1} = \frac{n+2}{2}. \] Thus, the correct answer is: \[ \boxed{\frac{n+2}{2}}. \]