Sequences & Series
GP and AP relations
Grade None
Question:
<p>If \(a, G, b\) are in GP and \(\dfrac{1}{a}, M, \dfrac{1}{b}\) are in AP, then \(M\) equals:</p>
<p>\(\dfrac{2ab}{a+b}\)</p>
<p>\(\dfrac{a+b}{2ab}\)</p>
<p>\(\dfrac{a+b}{2}\)</p>
<p>\(\sqrt{ab}\)</p>
Step-by-Step Solution
Key Concept: Use the geometric mean property for GP (G² = ab) and the arithmetic mean property for AP (M = average of extremes). Then establish the relationship between a, b, and M using these conditions simultaneously.
<p><strong>Step 1:</strong> Since a, G, b are in GP, we have:</p><p>G² = ab → G = √(ab)</p><p><strong>Step 2:</strong> Since 1/a, M, 1/b are in AP, the middle term equals the arithmetic mean:</p><p>M = (1/a + 1/b)/2 = (a + b)/(2ab)</p><p><strong>Step 3:</strong> Simplify M in terms of a and b:</p><p>M = (a + b)/(2ab)</p><p><strong>Step 4:</strong> We can verify this is the answer. Notice that:</p><p>• If a = 1, b = 4: G = 2, and M = (1+4)/(2·1·4) = 5/8 = 2.5/(2·2) ✓</p><p>• M represents the relationship between the harmonic and geometric properties of a and b</p><p>∴ Answer: <strong>M = (a + b)/(2ab)</strong></p>
Correct Answer: A