Question

If one AM 'A' and two GM \( p \) and \( q \) are inserted between two given numbers, then find the value of \[ \frac{p^2}{q} + \frac{q^2}{p} \]

• A. A
• B. 2A
• C. 3A
• D. 4A

Hint:

In a set of AM and GM, the relationship between the means can simplify the calculation of expressions involving \( p \) and \( q \).

Answer 1

The Correct Option is B

Step 1: Apply the properties of AM and GM.
The arithmetic mean (AM) and geometric mean (GM) are related by the formula \( A = \frac{p + q}{2} \), and \( p \) and \( q \) are the geometric means between two numbers.
Step 2: Conclusion.
After applying the relations, we find that \( \frac{p^2}{q} + \frac{q^2}{p} = 2A \).
Final Answer: \[ \boxed{2A} \]