If `n_(1) and n_(2)` are the sizes, `G_(1) and G_(2)` the geometric means of two series respectively, then which one of the following expresses the geometric mean (G) of the combined series?

Options

A. `logG=(n_(1)G_(1)+n_(2)G_(2))/(n_(1)+n_(2))`
B. `logG=(n_(2)log G_(1)+n_(1)logG_(2))/(n_(1)+n_(2))`
C. `G=(n_(1)logG_(1)+n_(2)logG_(2))/(n_(1)+n_(2))`
D. None of the above

Explanation

To find the geometric mean \( G \) of the combined series given the sizes \( n_1 \) and \( n_2 \) and the geometric means \( G_1 \) and \( G_2 \) of the two series, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometric Mean**:
The geometric mean \( G \) of a series of numbers \( x_1, x_2, \ldots, x_n \) is given by:
\[ G = (x_1 \cdot x_2 \cdot \ldots \cdot x_n)^{\frac{1}{n}} \]
where \( n \) is the number of terms in the series.

2. **Apply the Definition to Each Series**:
For the first series with \( n_1 \) terms and geometric mean \( G_1 \):
\[ G_1 = (x_1 \cdot x_2 \cdot \ldots \cdot x_{n_1})^{\frac{1}{n_1}} \]
For the second series with \( n_2 \) terms and geometric mean \( G_2 \):
\[ G_2 = (y_1 \cdot y_2 \cdot \ldots \cdot y_{n_2})^{\frac{1}{n_2}} \]

3. **Combine the Two Series**:
The combined series will have \( n_1 + n_2 \) terms, which includes all terms from both series. The product of all terms in the combined series can be expressed as:
\[ \text{Combined Product} = (x_1 \cdot x_2 \cdot \ldots \cdot x_{n_1}) \cdot (y_1 \cdot y_2 \cdot \ldots \cdot y_{n_2}) = G_1^{n_1} \cdot G_2^{n_2} \]

4. **Calculate the Geometric Mean of the Combined Series**:
The geometric mean \( G \) of the combined series is then given by:
\[ G = \left(G_1^{n_1} \cdot G_2^{n_2}\right)^{\frac{1}{n_1 + n_2}} \]

5. **Simplify the Expression**:
Using properties of exponents, we can simplify this to:
\[ G = G_1^{\frac{n_1}{n_1 + n_2}} \cdot G_2^{\frac{n_2}{n_1 + n_2}} \]

### Final Result:
Thus, the geometric mean \( G \) of the combined series is expressed as:
\[ G = G_1^{\frac{n_1}{n_1 + n_2}} \cdot G_2^{\frac{n_2}{n_1 + n_2}} \]