Geometric Progression (GP) and Arithmetic Mean (AM) Problem

The question asks for the value of the expression $\frac{g_1^2}{g_2} + \frac{g_2^2}{g_1}$, given that p, $g_1$, $g_2$, and q are terms in a Geometric Progression (GP), and m is the Arithmetic Mean (AM) of p and q.

GP Properties Analysis

Let the first term of the GP be p and the common ratio be r.

• Since p, $g_1$, $g_2$, q are in GP, we can express the terms as:
  • $g_1 = p \cdot r$
  • $g_2 = p \cdot r^2$
  • $q = p \cdot r^3$

Simplifying the Expression

Now, substitute these terms into the expression $\frac{g_1^2}{g_2} + \frac{g_2^2}{g_1}$:

• Calculate the first part: $ \frac{g_1^2}{g_2} = \frac{(p \cdot r)^2}{p \cdot r^2} = \frac{p^2 \cdot r^2}{p \cdot r^2} = p $
• Calculate the second part: $ \frac{g_2^2}{g_1} = \frac{(p \cdot r^2)^2}{p \cdot r} = \frac{p^2 \cdot r^4}{p \cdot r} = p \cdot r^3 $
• Recognize that $p \cdot r^3$ is equal to the fourth term, q.
• So, the expression simplifies to: $ \frac{g_1^2}{g_2} + \frac{g_2^2}{g_1} = p + q $

Relating to Arithmetic Mean (AM)

The problem states that m is the Arithmetic Mean (AM) of p and q.

• By definition of AM: $ m = \frac{p+q}{2} $
• Rearranging this formula gives: $ p+q = 2m $

Final Calculation

Substitute the value of $p+q$ back into the simplified expression:

• $ \frac{g_1^2}{g_2} + \frac{g_2^2}{g_1} = p + q = 2m $

Therefore, the value of the expression is $2m$.