If the first item is increased by 1, second by 2 and so on, then the new mean is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Define the Original Observations**:
Let the original observations be \( x_1, x_2, \ldots, x_n \).

2. **Calculate the Original Mean**:
The mean \( \bar{x} \) of these observations is given by:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]

3. **Define the New Observations**:
The new observations after the specified increases will be:
\[ \text{New observations: } (x_1 + 1), (x_2 + 2), \ldots, (x_n + n) \]

4. **Calculate the Sum of New Observations**:
The sum of the new observations can be expressed as:
\[ (x_1 + 1) + (x_2 + 2) + \ldots + (x_n + n) = (x_1 + x_2 + \ldots + x_n) + (1 + 2 + \ldots + n) \]
The sum of the first \( n \) natural numbers is given by:
\[ 1 + 2 + \ldots + n = \frac{n(n + 1)}{2} \]
Therefore, the sum of the new observations becomes:
\[ \text{Sum of new observations} = (x_1 + x_2 + \ldots + x_n) + \frac{n(n + 1)}{2} \]

5. **Calculate the New Mean**:
The new mean \( \bar{x}_{new} \) is given by:
\[ \bar{x}_{new} = \frac{\text{Sum of new observations}}{n} \]
Substituting the sum we found:
\[ \bar{x}_{new} = \frac{(x_1 + x_2 + \ldots + x_n) + \frac{n(n + 1)}{2}}{n} \]
This can be rewritten as:
\[ \bar{x}_{new} = \frac{n \bar{x} + \frac{n(n + 1)}{2}}{n} \]
Simplifying further:
\[ \bar{x}_{new} = \bar{x} + \frac{n + 1}{2} \]

### Final Result:
Thus, the new mean after the specified increases is:
\[ \bar{x}_{new} = \bar{x} + \frac{n + 1}{2} \]