If `x_(i)gt0,y_(i)gt0(i=1,2,3,......n)` are the values of two variable X and Y with geometric mean P and Q respectively, then the geometric mean of `(X)/(Y)` is

Options

A. `(P)/(Q)`
B. antilog`((P)/(Q))`
C. `n(logP-logQ)`
D. `n(logP+logQ)`

Explanation

To solve the problem, we need to find the geometric mean of the ratio \( \frac{X}{Y} \) given that \( x_i > 0 \) and \( y_i > 0 \) for \( i = 1, 2, 3, \ldots, n \) with geometric means \( P \) and \( Q \) respectively.

### Step-by-Step Solution:

1. **Definition of Geometric Mean**:
The geometric mean \( P \) of the values \( x_1, x_2, \ldots, x_n \) is defined as:
\[ P = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} \]
Similarly, the geometric mean \( Q \) of the values \( y_1, y_2, \ldots, y_n \) is defined as:
\[ Q = \sqrt[n]{y_1 \cdot y_2 \cdot \ldots \cdot y_n} \]

2. **Finding the Geometric Mean of \( \frac{X}{Y} \)**:
We want to find the geometric mean of the ratios \( \frac{x_i}{y_i} \) for \( i = 1, 2, \ldots, n \). The geometric mean of \( \frac{x_1}{y_1}, \frac{x_2}{y_2}, \ldots, \frac{x_n}{y_n} \) is given by:
\[ G = \sqrt[n]{\frac{x_1}{y_1} \cdot \frac{x_2}{y_2} \cdot \ldots \cdot \frac{x_n}{y_n}} \]

3. **Simplifying the Expression**:
We can rewrite the expression for \( G \):
\[ G = \sqrt[n]{\frac{x_1 \cdot x_2 \cdot \ldots \cdot x_n}{y_1 \cdot y_2 \cdot \ldots \cdot y_n}} = \frac{\sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n}}{\sqrt[n]{y_1 \cdot y_2 \cdot \ldots \cdot y_n}} \]

4. **Substituting the Geometric Means**:
From the definitions of \( P \) and \( Q \), we can substitute:
\[ G = \frac{P}{Q} \]

5. **Final Result**:
Therefore, the geometric mean of \( \frac{X}{Y} \) is:
\[ G = \frac{P}{Q} \]

### Summary:
The geometric mean of the ratios \( \frac{x_i}{y_i} \) is equal to the ratio of the geometric means \( P \) and \( Q \) of the sequences \( x_i \) and \( y_i \) respectively.