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 \(2m\).
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