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 identities for sums of cubes and the relationship between AM and GM to simplify such expressions.

Answer 1

The Correct Option is C

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 Identity for Sum of Cubes 
We are asked to find the value of \( p^3 + q^3 \). 
Since \( p = q \), we can use the identity for the sum of cubes: \[ p^3 + q^3 = (p + q)(p^2 - pq + q^2). \] Since \( p = q \), this simplifies to: \[ p^3 + q^3 = 2p^3. \] Step 4: Express \( p^3 \) in Terms of \( a \) and \( xy \) 
From the AM-GM inequality, we know that \( p = \sqrt{xy} \). So, we compute \( p^3 \): \[ p^3 = (\sqrt{xy})^3 = (xy)^{3/2}. \] Step 5: Final Expression 
We now have the expression for \( p^3 + q^3 \), which is: \[ p^3 + q^3 = \frac{2pq}{a}. \] Thus, the correct answer is: \[ \boxed{\frac{2pq}{a}}. \]