If each of n numbers `x_(i)=i`, is replaced by `(i+1)x_(i)`, then the new mean is

Options

A. `((n+1)(n+2))/(n)`
B. `n+1`
C. `((n+1)(n+2))/(3)`
D. none of these

Answer

To solve the problem, we need to find the new mean after replacing each of the n numbers \( x_i = i \) with \( y_i = (i + 1)x_i \).

### Step-by-Step Solution:

1. **Define the Original Series**:
The original series consists of \( n \) numbers where \( x_i = i \). Thus, the numbers are:
\[ x_1 = 1, x_2 = 2, x_3 = 3, \ldots, x_n = n \]

2. **Define the New Series**:
Each number \( x_i \) is replaced by \( y_i = (i + 1)x_i \). Substituting \( x_i \) gives:
\[ y_i = (i + 1)i = i^2 + i \]
Therefore, the new series is:
\[ y_1 = 1^2 + 1 = 2, \quad y_2 = 2^2 + 2 = 6, \quad y_3 = 3^2 + 3 = 12, \ldots, \quad y_n = n^2 + n \]

3. **Calculate the New Mean**:
The mean of the new series is given by the formula:
\[ \text{Mean} = \frac{\sum_{i=1}^{n} y_i}{n} \]
We need to calculate \( \sum_{i=1}^{n} y_i \):
\[ \sum_{i=1}^{n} y_i = \sum_{i=1}^{n} (i^2 + i) = \sum_{i=1}^{n} i^2 + \sum_{i=1}^{n} i \]

4. **Use the Formulas for Sums**:
The formulas for the sums are:
- Sum of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} i = \frac{n(n + 1)}{2} \]
- Sum of the squares of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} \]

5. **Substitute the Sums**:
Now substituting these formulas into our expression for \( \sum_{i=1}^{n} y_i \):
\[ \sum_{i=1}^{n} y_i = \frac{n(n + 1)(2n + 1)}{6} + \frac{n(n + 1)}{2} \]

6. **Combine the Terms**:
To combine these fractions, we need a common denominator:
\[ \frac{n(n + 1)(2n + 1)}{6} + \frac{3n(n + 1)}{6} = \frac{n(n + 1)(2n + 1 + 3)}{6} = \frac{n(n + 1)(2n + 4)}{6} \]

7. **Simplify**:
This simplifies to:
\[ \sum_{i=1}^{n} y_i = \frac{n(n + 1)(2(n + 2))}{6} = \frac{n(n + 1)(n + 2)}{3} \]

8. **Calculate the Mean**:
Now, substituting back into the mean formula:
\[ \text{Mean} = \frac{\sum_{i=1}^{n} y_i}{n} = \frac{\frac{n(n + 1)(n + 2)}{3}}{n} = \frac{(n + 1)(n + 2)}{3} \]

### Final Answer:
Thus, the new mean is:
\[ \text{New Mean} = \frac{(n + 1)(n + 2)}{3} \]