<p>In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals</p>
<p>\(\dfrac{1}{2}(1 - \sqrt{5})\)</p>
<p>\(\dfrac{1}{2}\sqrt{5}\)</p>
<p>\(\sqrt{5}\)</p>
<p>\(\dfrac{1}{2}(\sqrt{5} - 1)\)</p>
Step-by-Step Solution
Key Concept: If each term equals the sum of the next two terms in a GP with first term a and common ratio r, then a = ar + ar², which gives a relationship that r must satisfy independent of a. Rearranging yields a quadratic in r.
<p><strong>Step 1:</strong> Let the GP be a, ar, ar², ar³, ... with a > 0 and common ratio r > 0 (given positive terms).</p><p><strong>Step 2:</strong> Apply the condition that each term equals the sum of the next two terms:</p><p>ar = ar² + ar³</p><p><strong>Step 3:</strong> Divide both sides by ar (since a ≠ 0, r ≠ 0):</p><p>1 = r + r²</p><p>r² + r - 1 = 0</p><p><strong>Step 4:</strong> Using the quadratic formula:</p><p>r = (-1 ± √(1 + 4))/2 = (-1 ± √5)/2</p><p><strong>Step 5:</strong> Since we need r > 0 (positive terms in GP):</p><p>r = (-1 + √5)/2 = (√5 - 1)/2 ≈ 0.618</p><p>This is the reciprocal of the golden ratio φ = (1 + √5)/2.</p><p>∴ Answer: <strong>r = (√5 - 1)/2</strong></p>
Correct Answer: D