The mean of n observation is `barX`. If the first observation is increased by `1^(2)`, second by `2^(2)` and so on, then the new mean is

Options

A. `barX+n((n+1)(2n+1))/(6)`
B. `barX+((n+1)(2n+1))/(6)`
C. `barX+(n+1)/(2)`
D. `barX+((n+1))/((4)`

Answer

To find the new mean after increasing each observation by the squares of their respective indices, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Observations and Mean**:
Let the n observations be \( x_1, x_2, x_3, \ldots, x_n \). The mean of these observations is given by:
\[ \bar{X} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

2. **Formulate the New Observations**:
According to the problem, the first observation is increased by \( 1^2 \), the second by \( 2^2 \), and so on until the nth observation, which is increased by \( n^2 \). Therefore, the new observations \( y_1, y_2, \ldots, y_n \) can be expressed as:
\[ y_1 = x_1 + 1^2, \quad y_2 = x_2 + 2^2, \quad y_3 = x_3 + 3^2, \ldots, \quad y_n = x_n + n^2 \]

3. **Calculate the New Mean**:
The new mean \( \bar{Y} \) can be calculated as:
\[ \bar{Y} = \frac{y_1 + y_2 + y_3 + \ldots + y_n}{n} \]
Substituting the expressions for \( y_i \):
\[ \bar{Y} = \frac{(x_1 + 1^2) + (x_2 + 2^2) + (x_3 + 3^2) + \ldots + (x_n + n^2)}{n} \]

4. **Simplify the Expression**:
This can be simplified as:
\[ \bar{Y} = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) + (1^2 + 2^2 + 3^2 + \ldots + n^2)}{n} \]
We know that \( x_1 + x_2 + \ldots + x_n = n \bar{X} \), so we can substitute this into the equation:
\[ \bar{Y} = \frac{n \bar{X} + (1^2 + 2^2 + 3^2 + \ldots + n^2)}{n} \]

5. **Sum of Squares Formula**:
The sum of the squares of the first n natural numbers is given by:
\[ 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \]
Substituting this into the equation for \( \bar{Y} \):
\[ \bar{Y} = \frac{n \bar{X} + \frac{n(n + 1)(2n + 1)}{6}}{n} \]

6. **Final Expression**:
This simplifies to:
\[ \bar{Y} = \bar{X} + \frac{(n + 1)(2n + 1)}{6} \]

### Conclusion:
Thus, the new mean after the specified increases is:
\[ \bar{Y} = \bar{X} + \frac{(n + 1)(2n + 1)}{6} \]