Question

If one AM (Arithmetic mean) \( a \) and two GM's (Geometric means) \( p \) and \( q \) be inserted between any two positive numbers, the value of \( p^3 + q^3 \) is:

• A. \( 2a \, pq \)
• B. \( \frac{pq}{a} \)
• C. \( \frac{2pq}{a} \)
• D. \( p + q + a \)

Hint:

Use the relationships between AM and GM and algebraic identities to simplify expressions involving sums of cubes.

Answer 1

The Correct Option is A

Let the two numbers be \( x \) and \( y \). The arithmetic mean and geometric means between \( x \) and \( y \) are defined as follows: 
Step 1: Arithmetic Mean (AM) Formula 
The arithmetic mean (AM) of two numbers \( x \) and \( y \) is given by: \[ a = \frac{x + y}{2}. \] Step 2: Geometric Means (GM) Formula 
The geometric means inserted between \( x \) and \( y \) are \( p \) and \( q \), and the formula for the geometric mean between two numbers is: \[ p = \sqrt{xy}, \quad q = \sqrt{xy}. \] Step 3: Use of Arithmetic Mean and Geometric Mean Relationship Since \( p \) and \( q \) are geometric means, we know the relationship between the AM and GM: \[ a = \frac{x + y}{2}, \quad p = q = \sqrt{xy}. \] Step 4: Compute \( p^3 + q^3 \) 
We are asked to find \( p^3 + q^3 \). 
Using the identity for the sum of cubes: \[ p^3 + q^3 = (p + q)(p^2 - pq + q^2). \] Since \( p = q \), this simplifies to: \[ p^3 + q^3 = 2p^3. \] Now, express \( p^3 \) in terms of \( a \) and \( xy \). Using the relation between AM and GM, we get: \[ p^3 = 2a \, pq. \] Thus, the correct answer is: \[ \boxed{2a \, pq}. \]