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 denoted as \( a_1, a_2, ..., a_n \).The mean (\(\bar{X}\)) is calculated as:\[\bar{X} = \frac{a_1 + a_2 + ... + a_n}{n}\]Given a new condition where each item \( a_i \) is augmented by \( i \):\[\bar{X}_{{new}} = \frac{(a_1+1) + (a_2+2) + ... + (a_n+n)}{n}\]Applying the formula for the sum of the first \( n \) natural numbers, the new mean (\(\bar{X}_{{new}}\)) can be expressed in terms of the original mean (\(\bar{X}\)) as follows:\[\bar{X}_{{new}} = \bar{X} + \frac{n(n+1)}{2n} = \bar{X} + \frac{n+1}{2}\]