To solve the problem, we need to find the mean of the given set of numbers: \( ax_1, ax_2, \ldots, ax_n, \frac{x_1}{a}, \frac{x_2}{a}, \ldots, \frac{x_n}{a} \).
### Step-by-Step Solution:
1. **Understand the Mean**:
The mean \( \bar{x} \) of the values \( x_1, x_2, \ldots, x_n \) is given by:
\[
\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}
\]
2. **Calculate the Mean of the New Set**:
We need to find the mean of the new set of numbers:
\[
ax_1, ax_2, \ldots, ax_n, \frac{x_1}{a}, \frac{x_2}{a}, \ldots, \frac{x_n}{a}
\]
This set has a total of \( 2n \) observations.
3. **Sum of the New Set**:
The sum of the first \( n \) terms (i.e., \( ax_1, ax_2, \ldots, ax_n \)) is:
\[
a(x_1 + x_2 + \ldots + x_n) = a \cdot n \bar{x}
\]
The sum of the second \( n \) terms (i.e., \( \frac{x_1}{a}, \frac{x_2}{a}, \ldots, \frac{x_n}{a} \)) is:
\[
\frac{1}{a}(x_1 + x_2 + \ldots + x_n) = \frac{n \bar{x}}{a}
\]
4. **Total Sum**:
Now, we can combine these two sums:
\[
\text{Total Sum} = a \cdot n \bar{x} + \frac{n \bar{x}}{a}
\]
5. **Mean Calculation**:
The mean of the new set is given by the total sum divided by the total number of observations:
\[
\text{Mean} = \frac{a \cdot n \bar{x} + \frac{n \bar{x}}{a}}{2n}
\]
6. **Factor Out \( n \bar{x} \)**:
We can factor out \( n \bar{x} \):
\[
\text{Mean} = \frac{n \bar{x} \left( a + \frac{1}{a} \right)}{2n}
\]
7. **Simplify**:
The \( n \) cancels out:
\[
\text{Mean} = \frac{\bar{x} \left( a + \frac{1}{a} \right)}{2}
\]
### Final Answer:
Thus, the mean of the set \( ax_1, ax_2, \ldots, ax_n, \frac{x_1}{a}, \frac{x_2}{a}, \ldots, \frac{x_n}{a} \) is:
\[
\frac{\bar{x} \left( a + \frac{1}{a} \right)}{2}
\]
