<p>Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is</p>
Step-by-Step Solution
Key Concept: Set up the G.P. as a/r, a, ar (where r > 1 for increasing sequence), then use the A.P. condition on the transformed terms to create an equation in r.
<p><strong>Step 1:</strong> Let the three terms of the G.P. be a/r, a, ar where a > 0 and r > 1 (increasing G.P.)</p><p><strong>Step 2:</strong> When the middle term is doubled, the new sequence is: a/r, 2a, ar</p><p><strong>Step 3:</strong> For these to be in A.P., the common difference must be constant:<br>2a - a/r = ar - 2a</p><p><strong>Step 4:</strong> Simplifying the left side: 2a - a/r = a(2 - 1/r) = a(2r - 1)/r<br>Simplifying the right side: ar - 2a = a(r - 2)</p><p><strong>Step 5:</strong> Equating: a(2r - 1)/r = a(r - 2)<br>Dividing by a: (2r - 1)/r = r - 2<br>2r - 1 = r(r - 2)<br>2r - 1 = r² - 2r<br>r² - 4r + 1 = 0</p><p><strong>Step 6:</strong> Using the quadratic formula: r = (4 ± √(16 - 4))/2 = (4 ± √12)/2 = (4 ± 2√3)/2 = 2 ± √3</p><p><strong>Step 7:</strong> Since r > 1 (increasing sequence), r = 2 + √3 ≈ 3.73 (valid) and r = 2 - √3 ≈ 0.27 (reject as r < 1)</p><p>∴ Answer: <strong>r = 2 + √3</strong></p>
Correct Answer: B