Problem:

If one Arithmetic Mean (AM) \( a \) and two Geometric Means \( p \) and \( q \) are inserted between any two positive numbers, find the value of: \[ p^3 + q^3 \]

Given:

• Let two positive numbers be \( A \) and \( B \).
• One AM: \( a = \frac{A + B}{2} \)
• Two GMs inserted: so the four terms in G.P. are: \[ A, \ p = \sqrt[3]{A^2B}, \ q = \sqrt[3]{AB^2}, \ B \]

Now calculate:

\[ pq = \sqrt[3]{A^2B} \cdot \sqrt[3]{AB^2} = \sqrt[3]{A^3B^3} = AB \]
\[ p^3 = A^2B, \quad q^3 = AB^2 \]
\[ p^3 + q^3 = A^2B + AB^2 = AB(A + B) \]

Also,

\[ 2apq = 2 \cdot \frac{A + B}{2} \cdot AB = AB(A + B) \]

✅ Therefore,

\( \boxed{p^3 + q^3 = 2apq} \)