The mean of `x_(1)+x_(2)+…+x_(n)` is M. When `x_(i) ,i=1,2,….,10,` is replaced by `x_(i)+10`, the mean is `M_(1)`. Then `M_(1)` =

Options

A. M
B. `M+10`
C. 10 M
D. `M+100`

Answer

To solve the problem step by step, we need to understand the concept of mean and how it changes when we modify the values in the dataset.

### Step-by-Step Solution:

1. **Understanding the Mean**:
The mean of a set of numbers \( x_1, x_2, \ldots, x_n \) is given by the formula:
\[ M = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
Here, \( n \) is the total number of terms. In this case, we have \( n = 10 \) and the mean is \( M \).

2. **Replacing Values**:
We are told that each \( x_i \) (where \( i = 1, 2, \ldots, 10 \)) is replaced by \( x_i + 10 \). Therefore, the new values become:
\[ x_1 + 10, x_2 + 10, \ldots, x_{10} + 10 \]

3. **Calculating the New Mean**:
The new mean \( M_1 \) can be calculated as:
\[ M_1 = \frac{(x_1 + 10) + (x_2 + 10) + \ldots + (x_{10} + 10)}{10} \]

4. **Simplifying the New Mean**:
We can simplify the expression for \( M_1 \):
\[ M_1 = \frac{(x_1 + x_2 + \ldots + x_{10}) + (10 \times 10)}{10} \]
Here, \( 10 \times 10 \) accounts for the 10 added to each of the 10 terms.

5. **Substituting the Original Mean**:
We know from the original mean \( M \):
\[ x_1 + x_2 + \ldots + x_{10} = 10M \]
Therefore, substituting this into the equation for \( M_1 \):
\[ M_1 = \frac{10M + 100}{10} \]

6. **Final Calculation**:
Simplifying the above expression:
\[ M_1 = M + 10 \]

### Conclusion:
The new mean \( M_1 \) after replacing each \( x_i \) with \( x_i + 10 \) is:
\[ M_1 = M + 10 \]