The mean of 'n' observations is `tildex`. If the first observation is increased by 1. The second by 2, the third by 3 and so on, then the new mean is equal to:

Options

A. `tildex+(2n+1)`
B. `tildex+((n+1))/2`
C. `tildex+(n+1)`
D. `tildex-((n-1))/2`

Answer

To find the new mean after increasing the first observation by 1, the second by 2, the third by 3, and so on, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Mean**:
The mean of 'n' observations is given as \( \tilde{x} \). This can be expressed mathematically as:
\[ \tilde{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
where \( x_1, x_2, \ldots, x_n \) are the observations.

**Hint**: Recall that the mean is the sum of observations divided by the number of observations.

2. **Increase Each Observation**:
We increase the first observation \( x_1 \) by 1, the second observation \( x_2 \) by 2, the third observation \( x_3 \) by 3, and so forth, up to the nth observation \( x_n \) which is increased by \( n \). Thus, the new observations become:
\[ x_1 + 1, \quad x_2 + 2, \quad x_3 + 3, \quad \ldots, \quad x_n + n \]

**Hint**: Write down the new values of the observations clearly to avoid confusion.

3. **Calculate the New Sum**:
The new sum of the observations can be calculated as:
\[ (x_1 + 1) + (x_2 + 2) + (x_3 + 3) + \ldots + (x_n + n) = (x_1 + x_2 + x_3 + \ldots + x_n) + (1 + 2 + 3 + \ldots + n) \]
The sum of the first \( n \) natural numbers \( 1 + 2 + 3 + \ldots + n \) can be calculated using the formula:
\[ \frac{n(n + 1)}{2} \]
Therefore, the new sum becomes:
\[ \text{New Sum} = \sum_{i=1}^{n} x_i + \frac{n(n + 1)}{2} \]

**Hint**: Remember the formula for the sum of the first \( n \) natural numbers.

4. **Calculate the New Mean**:
The new mean \( \tilde{x}_{new} \) can now be calculated as:
\[ \tilde{x}_{new} = \frac{\text{New Sum}}{n} = \frac{\sum_{i=1}^{n} x_i + \frac{n(n + 1)}{2}}{n} \]
Substituting \( \sum_{i=1}^{n} x_i = n \tilde{x} \):
\[ \tilde{x}_{new} = \frac{n \tilde{x} + \frac{n(n + 1)}{2}}{n} = \tilde{x} + \frac{n + 1}{2} \]

**Hint**: Simplify the expression carefully to find the new mean.

### Final Result:
The new mean after the adjustments is:
\[ \tilde{x}_{new} = \tilde{x} + \frac{n + 1}{2} \]