If the mean of the set of numbers `x_1,x_2, x_3, ..., x_n` is `barx,` then the mean of the numbers `x_i+2i, 1 lt= i lt= n` is

Options

A. `overline (x)+2n`
B. `overline(x)+n+1`
C. `overline(x)+2`
D. `overline(x)+n`

Explanation

To find the mean of the numbers of the form \( y_i = x_i + 2i \) for \( 1 \leq i \leq n \), we can follow these steps:

### Step 1: Understand the given information
We know that the mean of the set of numbers \( x_1, x_2, \ldots, x_n \) is given by \( \bar{x} \). This can be expressed mathematically as:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]

### Step 2: Express the new set of numbers
The new set of numbers is defined as:
\[ y_i = x_i + 2i \]
for \( i = 1, 2, \ldots, n \).

### Step 3: Calculate the mean of the new set
The mean of the new numbers \( y_i \) can be calculated as:
\[ \bar{y} = \frac{\sum_{i=1}^{n} y_i}{n} \]
Substituting \( y_i \) into the equation:
\[ \bar{y} = \frac{\sum_{i=1}^{n} (x_i + 2i)}{n} \]

### Step 4: Separate the summation
We can separate the summation into two parts:
\[ \bar{y} = \frac{\sum_{i=1}^{n} x_i + \sum_{i=1}^{n} 2i}{n} \]
This simplifies to:
\[ \bar{y} = \frac{\sum_{i=1}^{n} x_i}{n} + \frac{\sum_{i=1}^{n} 2i}{n} \]

### Step 5: Substitute the mean of \( x_i \)
We know that:
\[ \frac{\sum_{i=1}^{n} x_i}{n} = \bar{x} \]
So we can write:
\[ \bar{y} = \bar{x} + \frac{\sum_{i=1}^{n} 2i}{n} \]

### Step 6: Calculate the sum \( \sum_{i=1}^{n} i \)
The sum of the first \( n \) natural numbers is given by the formula:
\[ \sum_{i=1}^{n} i = \frac{n(n + 1)}{2} \]
Thus,
\[ \sum_{i=1}^{n} 2i = 2 \cdot \frac{n(n + 1)}{2} = n(n + 1) \]

### Step 7: Substitute back into the mean equation
Now substituting this back into our mean equation:
\[ \bar{y} = \bar{x} + \frac{n(n + 1)}{n} = \bar{x} + (n + 1) \]

### Final Result
Thus, the mean of the numbers \( y_i = x_i + 2i \) is:
\[ \bar{y} = \bar{x} + (n + 1) \]