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

Options

A. `overline(X) +n`
B. `overline(X)+(n)/(2)`
C. `overline(X)+(n+1)/(2)`
D. None of these

Explanation

To find the new mean after increasing the first item by 1, the second by 2, and so on, we can follow these steps:

### Step 1: Understand the original mean
The mean of n items is given as \(\overline{X}\). This can be expressed mathematically as:
\[ \overline{X} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
From this, we can derive that:
\[ \overline{X} \cdot n = x_1 + x_2 + x_3 + \ldots + x_n \quad \text{(1)} \]

### Step 2: Calculate the new total after the increases
The new total after increasing the first item by 1, the second by 2, and so forth, can be expressed as:
\[ \text{New Total} = (x_1 + 1) + (x_2 + 2) + (x_3 + 3) + \ldots + (x_n + n) \]
This can be rewritten as:
\[ \text{New Total} = (x_1 + x_2 + x_3 + \ldots + x_n) + (1 + 2 + 3 + \ldots + n \]
Using equation (1), we can substitute the sum of the original items:
\[ \text{New Total} = \overline{X} \cdot n + (1 + 2 + 3 + \ldots + n) \]

### Step 3: Use the formula for the sum of the first n natural numbers
The sum of the first n natural numbers is given by the formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Substituting this into our equation gives:
\[ \text{New Total} = \overline{X} \cdot n + \frac{n(n + 1)}{2} \]

### Step 4: Calculate the new mean
The new mean \(\overline{X}'\) can be calculated by dividing the new total by n:
\[ \overline{X}' = \frac{\text{New Total}}{n} = \frac{\overline{X} \cdot n + \frac{n(n + 1)}{2}}{n} \]
This simplifies to:
\[ \overline{X}' = \overline{X} + \frac{n + 1}{2} \]

### Final Result
Thus, the new mean after the adjustments is:
\[ \overline{X}' = \overline{X} + \frac{n + 1}{2} \]