Question:
<p>The mean of \(n\) items is \(\bar{X}\). If the first item is increased by 1, second by 2 and so on, then the new mean is</p>
<p>\(\bar{X} + n\)</p>
<p>\(\bar{X} + \dfrac{n}{2}\)</p>
<p>\(\bar{X} + \dfrac{n+1}{2}\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: When each item is increased by its position number (1st by 1, 2nd by 2, etc.), the mean increases by the average of the increments, which equals (1+2+...+n)/n = (n+1)/2.
<p><strong>Step 1:</strong> Original mean of n items is X̄, so the sum of all items = nX̄</p><p><strong>Step 2:</strong> Total increase in all items = 1 + 2 + 3 + ... + n = n(n+1)/2</p><p><strong>Step 3:</strong> New sum of items = nX̄ + n(n+1)/2</p><p><strong>Step 4:</strong> New mean = [nX̄ + n(n+1)/2]/n = X̄ + (n+1)/2</p><p>∴ Answer: <strong>X̄ + (n+1)/2</strong> or equivalently <strong>X̄ + n/2 + 1/2</strong> (Option C)</p>
Correct Answer: C