If the arithmetic mean of the numbers `x_1, x_2, x_3, ..., x_n` is `barx,` then the arithmetic mean of the numbers `ax_1 +b, ax_2 +b, ax_3 +b, ....,ax_n +b,` where a, b are two constants, would be

Options

A. `overline (x)`
B. `na overline(x)+nb`
C. `a overline(x)`
D. `a overline(x)+b`

Explanation

To find the arithmetic mean of the transformed numbers \( ax_1 + b, ax_2 + b, ax_3 + b, \ldots, ax_n + b \), we can follow these steps:

### Step 1: Understand the given arithmetic mean
We know that the arithmetic mean \( \bar{x} \) of the numbers \( x_1, x_2, x_3, \ldots, x_n \) is defined as:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

### Step 2: Write the sum of the original numbers
From the definition of the arithmetic mean, we can express the sum of the numbers as:
\[ x_1 + x_2 + x_3 + \ldots + x_n = n \bar{x} \]

### Step 3: Calculate the sum of the transformed numbers
Now, we need to calculate the sum of the transformed numbers \( ax_1 + b, ax_2 + b, ax_3 + b, \ldots, ax_n + b \):
\[ \text{Sum} = (ax_1 + b) + (ax_2 + b) + (ax_3 + b) + \ldots + (ax_n + b) \]
This can be simplified as:
\[ \text{Sum} = (ax_1 + ax_2 + ax_3 + \ldots + ax_n) + (b + b + b + \ldots + b) \]
The second part consists of \( n \) terms of \( b \), so it becomes \( nb \):
\[ \text{Sum} = a(x_1 + x_2 + x_3 + \ldots + x_n) + nb \]

### Step 4: Substitute the sum of original numbers
Now, substituting the sum of the original numbers from Step 2:
\[ \text{Sum} = a(n \bar{x}) + nb \]
This simplifies to:
\[ \text{Sum} = an \bar{x} + nb \]

### Step 5: Calculate the arithmetic mean of the transformed numbers
Now, we can find the arithmetic mean of the transformed numbers:
\[ \text{Arithmetic Mean} = \frac{\text{Sum}}{n} = \frac{an \bar{x} + nb}{n} \]
This can be simplified to:
\[ \text{Arithmetic Mean} = a \bar{x} + b \]

### Final Answer
Thus, the arithmetic mean of the numbers \( ax_1 + b, ax_2 + b, ax_3 + b, \ldots, ax_n + b \) is:
\[ \boxed{a \bar{x} + b} \]