If `g_(1)` and `g_(2)` be the geometric means of two series of `n_(1)` and `n_(2)` items. Then the G.M. of the series obtained on combining them is

Options

A. `[(g_(1))^(n_(1)).(g_(2))^(n_(2))]^((1)/(n_(1)+n_(2)))`
B. `(g_(1)g_(2))^((n_(1))/(n_(1)+n_(2)))`
C. `(g_(1)g_(2))^((n_(2))/(n_(1)+n_(2)))`
D. `(g_(1)g_(2))^((*n_(1)n_(2))/(n_(1)+n_(2)))`

Answer

To find the geometric mean (G.M.) of two combined series with geometric means \( G_1 \) and \( G_2 \) for \( n_1 \) and \( n_2 \) items respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Geometric Means**:
- The geometric mean \( G_1 \) of the first series with \( n_1 \) items is given by:
\[ G_1 = (A_1 \cdot A_2 \cdot A_3 \cdots A_{n_1})^{\frac{1}{n_1}} \]
- The geometric mean \( G_2 \) of the second series with \( n_2 \) items is given by:
\[ G_2 = (B_1 \cdot B_2 \cdot B_3 \cdots B_{n_2})^{\frac{1}{n_2}} \]

2. **Combining the Two Series**:
- When we combine both series, we have a total of \( n_1 + n_2 \) items. The combined series consists of all items from both series:
\[ A_1, A_2, \ldots, A_{n_1}, B_1, B_2, \ldots, B_{n_2} \]

3. **Finding the Geometric Mean of the Combined Series**:
- The geometric mean \( G \) of the combined series is given by:
\[ G = (A_1 \cdot A_2 \cdots A_{n_1} \cdot B_1 \cdot B_2 \cdots B_{n_2})^{\frac{1}{n_1 + n_2}} \]

4. **Expressing the Combined Product in Terms of \( G_1 \) and \( G_2 \)**:
- We can express the product of the first series in terms of \( G_1 \):
\[ A_1 \cdot A_2 \cdots A_{n_1} = G_1^{n_1} \]
- Similarly, for the second series:
\[ B_1 \cdot B_2 \cdots B_{n_2} = G_2^{n_2} \]

5. **Substituting Back into the Geometric Mean Formula**:
- Substituting these expressions into the formula for \( G \):
\[ G = (G_1^{n_1} \cdot G_2^{n_2})^{\frac{1}{n_1 + n_2}} \]

6. **Simplifying the Expression**:
- This can be simplified 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 of the series obtained by combining them is:
\[ G = G_1^{\frac{n_1}{n_1 + n_2}} \cdot G_2^{\frac{n_2}{n_1 + n_2}} \]