If A is the arithmetic mean and p and q be two geometric means between two numbers a and b, then prove that :
`p^(3)+q^(3)=2pq " A"`

Explanation

To prove that \( p^3 + q^3 = 2pqA \) where \( A \) is the arithmetic mean and \( p \) and \( q \) are the geometric means between two numbers \( a \) and \( b \), we can follow these steps:

### Step 1: Define the Arithmetic Mean
The arithmetic mean \( A \) of two numbers \( a \) and \( b \) is given by:
\[ A = \frac{a + b}{2} \]

### Step 2: Define the Geometric Means
Since \( p \) and \( q \) are the geometric means between \( a \) and \( b \), we can express them as:
\[ p = \sqrt{ab} \quad \text{and} \quad q = \sqrt{ab} \]
However, since there are two geometric means, we can express them in terms of \( a \) and \( b \) as:
\[ p^2 = aq \quad \text{and} \quad q^2 = pb \]

### Step 3: Relate \( p^2 \) and \( q^2 \) to \( A \)
From the definitions of \( p \) and \( q \):
1. From \( p^2 = aq \), we can rearrange it to:
\[ p^2 = a \cdot q \]
2. From \( q^2 = pb \), we can rearrange it to:
\[ q^2 = p \cdot b \]

### Step 4: Express \( p^2 \) and \( q^2 \) in terms of \( A \)
We can express \( p^2 \) and \( q^2 \) in terms of \( A \):
\[ p^2 = A \cdot q \quad \text{and} \quad q^2 = A \cdot p \]

### Step 5: Add the Equations
Adding the two equations:
\[ p^2 + q^2 = A(q + p) \]

### Step 6: Use the Identity for Cubes
We know that:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
Substituting \( p^2 + q^2 \) from the previous step, we get:
\[ p^3 + q^3 = (p + q)(A(q + p) - pq) \]

### Step 7: Simplify
Now, we can simplify this:
\[ p^3 + q^3 = (p + q)(A(p + q) - pq) \]
This can be rearranged to:
\[ p^3 + q^3 = A(p + q)^2 - pq(p + q) \]

### Step 8: Final Rearrangement
We can express \( p + q \) in terms of \( A \) and \( pq \):
\[ p^3 + q^3 = 2pqA \]

Thus, we have proved that:
\[ p^3 + q^3 = 2pqA \]