If the mean of a set of observations `x_(1),x_(2), …,x_(n)" is " bar(X)`, then the mean of the observations `x_(i) +2i , i=1, 2, ..., n` is

Options

A. `bar(X) +2`
B. `bar(X) +2n`
C. `bar(X)+(n+1)`
D. `X+n`

Answer

To find the mean of the observations \( x_i + 2i \) for \( i = 1, 2, \ldots, n \), we can follow these steps:

### Step 1: Understand the Mean of Original Observations
Given that the mean of the observations \( x_1, x_2, \ldots, x_n \) is \( \bar{X} \), we can express this mathematically:
\[ \bar{X} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]

### Step 2: Calculate the New Observations
The new observations are given by \( x_i + 2i \) for \( i = 1, 2, \ldots, n \). Therefore, we can write:
\[ \text{New Observations} = (x_1 + 2 \cdot 1), (x_2 + 2 \cdot 2), \ldots, (x_n + 2n) \]

### Step 3: Find the Sum of the New Observations
The sum of the new observations can be expressed as:
\[ \text{Sum} = (x_1 + 2 \cdot 1) + (x_2 + 2 \cdot 2) + \ldots + (x_n + 2n) \]
This can be simplified to:
\[ \text{Sum} = (x_1 + x_2 + \ldots + x_n) + 2(1 + 2 + \ldots + n) \]

### Step 4: Use the Formula for the Sum of the First n Natural Numbers
The sum of the first \( n \) natural numbers \( 1 + 2 + \ldots + n \) is given by the formula:
\[ 1 + 2 + \ldots + n = \frac{n(n + 1)}{2} \]
Thus, we can substitute this into our sum:
\[ \text{Sum} = (x_1 + x_2 + \ldots + x_n) + 2 \cdot \frac{n(n + 1)}{2} \]
This simplifies to:
\[ \text{Sum} = (x_1 + x_2 + \ldots + x_n) + n(n + 1) \]

### Step 5: Calculate the Mean of the New Observations
Now, we can find the mean of the new observations:
\[ \text{Mean} = \frac{\text{Sum}}{n} = \frac{(x_1 + x_2 + \ldots + x_n) + n(n + 1)}{n} \]
Substituting \( \bar{X} \) for the sum of the original observations:
\[ \text{Mean} = \frac{n \bar{X} + n(n + 1)}{n} \]
This simplifies to:
\[ \text{Mean} = \bar{X} + (n + 1) \]

### Final Answer
Thus, the mean of the observations \( x_i + 2i \) is:
\[ \bar{X} + (n + 1) \]