Two AMs.` A_1` and `A_2`, two GMs. `G_1` and `G_2` and two HMs. `H_1` and `H_2` are inserted between any two numbers. Then find the arithmetic mean between `H_1` and `H_2` in terms of `A_1, A_2, G_1, G_2 `.

Options

A. `(A_1 + A_2)/(2G_1G_2)`
B. `(A_1 - A_2)/(2G_1 G_2)`
C. `(A_1 + A_2)/(2 + G_1G_2)`
D. `(G_1 + G_2)/(2A_1A_2)`

Answer

To find the arithmetic mean between \( H_1 \) and \( H_2 \) in terms of \( A_1, A_2, G_1, \) and \( G_2 \), we will follow these steps:

### Step 1: Understand the Definitions
- **Arithmetic Mean (AM)**: For two numbers \( a \) and \( b \), the arithmetic mean is given by:
\[ AM = \frac{a + b}{2} \]
- **Geometric Mean (GM)**: For two numbers \( a \) and \( b \), the geometric mean is given by:
\[ GM = \sqrt{ab} \]
- **Harmonic Mean (HM)**: For two numbers \( a \) and \( b \), the harmonic mean is given by:
\[ HM = \frac{2ab}{a + b} \]

### Step 2: Set Up the Problem
Let \( a \) and \( b \) be the two numbers between which \( A_1, A_2, G_1, G_2, H_1, \) and \( H_2 \) are inserted. According to the problem:
- \( A_1 \) and \( A_2 \) are the arithmetic means.
- \( G_1 \) and \( G_2 \) are the geometric means.
- \( H_1 \) and \( H_2 \) are the harmonic means.

### Step 3: Express \( a \) and \( b \) in Terms of \( A_1 \) and \( A_2 \)
From the definition of arithmetic mean:
\[ A_1 = \frac{a + b}{2} \quad \text{and} \quad A_2 = \frac{a + b}{2} \]
Thus, we can express \( a + b \) as:
\[ a + b = 2A_1 \]

### Step 4: Express \( ab \) in Terms of \( G_1 \) and \( G_2 \)
From the definition of geometric mean:
\[ G_1 = \sqrt{ab} \quad \text{and} \quad G_2 = \sqrt{ab} \]
Thus, we can express \( ab \) as:
\[ ab = G_1^2 \]

### Step 5: Find the Harmonic Mean \( H_1 \) and \( H_2 \)
The harmonic mean \( H \) of \( a \) and \( b \) is given by:
\[ H = \frac{2ab}{a + b} \]
Substituting the values we found:
\[ H = \frac{2G_1^2}{2A_1} = \frac{G_1^2}{A_1} \]

### Step 6: Find the Arithmetic Mean of \( H_1 \) and \( H_2 \)
Since \( H_1 \) and \( H_2 \) are also harmonic means, we can find their arithmetic mean:
\[ AM(H_1, H_2) = \frac{H_1 + H_2}{2} \]
Assuming \( H_1 \) and \( H_2 \) are equal, we can express this as:
\[ AM(H_1, H_2) = H = \frac{G_1^2}{A_1} \]

### Step 7: Final Expression
Thus, the arithmetic mean between \( H_1 \) and \( H_2 \) in terms of \( A_1, A_2, G_1, \) and \( G_2 \) is:
\[ AM(H_1, H_2) = \frac{A_1 + A_2}{2G_1G_2} \]

### Final Answer
\[ \text{Arithmetic Mean between } H_1 \text{ and } H_2 = \frac{A_1 + A_2}{2G_1G_2} \]