GM of the numbers `3,3^2,3^3,...,3^n` is

Options

A. `3^(2//n)`
B. `3^((n-1)/(2))`
C. `3^(n//2)`
D. `3^((n+1)/(2))`

Explanation

To find the geometric mean (GM) of the numbers \(3, 3^2, 3^3, \ldots, 3^n\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Terms**:
The numbers we have are \(a_1 = 3\), \(a_2 = 3^2\), \(a_3 = 3^3\), ..., \(a_n = 3^n\).

2. **Write the Formula for GM**:
The formula for the geometric mean of \(n\) terms \(a_1, a_2, \ldots, a_n\) is given by:
\[ GM = (a_1 \cdot a_2 \cdot a_3 \cdots a_n)^{\frac{1}{n}} \]

3. **Calculate the Product of the Terms**:
We can express the product of the terms as:
\[ a_1 \cdot a_2 \cdot a_3 \cdots a_n = 3 \cdot 3^2 \cdot 3^3 \cdots 3^n \]
This can be simplified using the properties of exponents:
\[ = 3^{1 + 2 + 3 + \ldots + n} \]

4. **Sum of the Exponents**:
The sum \(1 + 2 + 3 + \ldots + n\) is a well-known formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]

5. **Substituting Back into the GM Formula**:
Now substituting the sum back into the product:
\[ a_1 \cdot a_2 \cdots a_n = 3^{\frac{n(n + 1)}{2}} \]

6. **Final Calculation of GM**:
Now we can find the GM:
\[ GM = \left(3^{\frac{n(n + 1)}{2}}\right)^{\frac{1}{n}} = 3^{\frac{n + 1}{2}} \]

### Final Answer:
Thus, the geometric mean of the numbers \(3, 3^2, 3^3, \ldots, 3^n\) is:
\[ GM = 3^{\frac{n + 1}{2}} \]