If ` g_(1) , g_(2) , g_(3)` are three geometric means between
two positive numbers a and b , then ` g_(1) g_(3)` is equal to

Options

A. `g_(2)`
B. `2g_(2)`
C. `g_(2)^(2)`
D. `g_(2)^(3)`

Answer

To solve the problem, we need to find the product of the first and third geometric means, \( g_1 \) and \( g_3 \), between two positive numbers \( a \) and \( b \).

### Step-by-Step Solution:

1. **Understanding Geometric Means**:
Given two positive numbers \( a \) and \( b \), if \( g_1, g_2, g_3 \) are three geometric means between them, we can express them in terms of \( a \) and \( b \) as follows:
\[ g_1 = a \cdot r, \quad g_2 = a \cdot r^2, \quad g_3 = a \cdot r^3 \]
where \( r \) is the common ratio.

2. **Finding the Common Ratio**:
Since \( g_1, g_2, g_3 \) are between \( a \) and \( b \), we can express \( b \) in terms of \( a \) and \( r \):
\[ g_1 \cdot g_2 \cdot g_3 = a \cdot r \cdot a \cdot r^2 \cdot a \cdot r^3 = a^3 \cdot r^6 \]
The geometric mean of \( a \) and \( b \) can also be expressed as:
\[ \sqrt[5]{a \cdot b} = a^{1/5} \cdot b^{4/5} \]
Since \( g_1, g_2, g_3 \) are geometric means, we can find \( r \) such that:
\[ g_1 \cdot g_2 \cdot g_3 = a \cdot b \]

3. **Calculating \( g_1 \cdot g_3 \)**:
Now, we want to find \( g_1 \cdot g_3 \):
\[ g_1 \cdot g_3 = (a \cdot r) \cdot (a \cdot r^3) = a^2 \cdot r^4 \]

4. **Finding \( r^4 \)**:
From the expression for \( g_2 \):
\[ g_2 = a \cdot r^2 \]
We can express \( r \) in terms of \( a \) and \( b \):
\[ r = \left( \frac{b}{a} \right)^{1/4} \]
Therefore,
\[ r^4 = \frac{b}{a} \]

5. **Substituting \( r^4 \) back into \( g_1 \cdot g_3 \)**:
Now substituting \( r^4 \) into the equation for \( g_1 \cdot g_3 \):
\[ g_1 \cdot g_3 = a^2 \cdot \frac{b}{a} = a \cdot b \]

### Final Result:
Thus, the product \( g_1 \cdot g_3 \) is equal to:
\[ g_1 \cdot g_3 = ab \]