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