Question

The mean of \( n \) items is \( X \). If the first item is increased by 1, second by 2, and so on, the new mean is:

• A. \( \bar{X} + \frac{x}{2} \)
• B. \( \bar{X} + x \)
• C. \( \bar{X} + \frac{n+1}{2} \)
• D. None of these

Hint:

Use summation formulas for sequences to simplify mean calculations.

Answer 1

The Correct Option is C

Let the items be \( a_1, a_2, ..., a_n \). \[ \bar{X} = \frac{a_1 + a_2 + ... + a_n}{n} \] Now, given the condition: \[ \bar{X}_{{new}} = \frac{(a_1+1) + (a_2+2) + ... + (a_n+n)}{n} \] Using the sum of the first \( n \) natural numbers: \[ \bar{X}_{{new}} = \bar{X} + \frac{n(n+1)}{2n} = \bar{X} + \frac{n+1}{2} \]

Answer 2

Step 1: Understanding the problem
We are given that the mean of \( n \) items is \( \bar{X} \). The first item is increased by 1, the second by 2, and so on. We need to determine the new mean after these changes.

Step 2: Formula for the original mean
The mean \( \bar{X} \) of the \( n \) items is given by: \[ \bar{X} = \frac{S}{n} \] where \( S \) is the sum of the original \( n \) items.

Step 3: New sum after changes
The first item is increased by 1, the second by 2, the third by 3, and so on. The total increase in the sum is: \[ 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \] Therefore, the new sum of the items is: \[ S_{\text{new}} = S + \frac{n(n+1)}{2} \] Step 4: New mean
The new mean is given by the new sum divided by \( n \): \[ \text{New Mean} = \frac{S_{\text{new}}}{n} = \frac{S + \frac{n(n+1)}{2}}{n} \] Substitute \( S = n \bar{X} \) (since \( \bar{X} \) is the original mean): \[ \text{New Mean} = \frac{n \bar{X} + \frac{n(n+1)}{2}}{n} \] Simplify the expression: \[ \text{New Mean} = \bar{X} + \frac{n+1}{2} \] Step 5: Final Answer
The new mean is:

\( \bar{X} + \frac{n+1}{2} \)