<p>If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is</p>
Step-by-Step Solution
Key Concept: Use the A.P. term formula to express the 2nd, 5th, and 9th terms, then apply the G.P. condition that the middle term squared equals the product of the other two terms.
<p><strong>Step 1:</strong> Let the A.P. have first term <em>a</em> and common difference <em>d</em>.</p><p>Then: a₂ = a + d, a₅ = a + 4d, a₉ = a + 8d</p><p><strong>Step 2:</strong> Since these are in G.P., apply the condition a₅² = a₂ · a₉:</p><p>(a + 4d)² = (a + d)(a + 8d)</p><p><strong>Step 3:</strong> Expand both sides:</p><p>a² + 8ad + 16d² = a² + 8ad + ad + 8d²</p><p>a² + 8ad + 16d² = a² + 9ad + 8d²</p><p><strong>Step 4:</strong> Simplify:</p><p>16d² - 8d² = 9ad - 8ad</p><p>8d² = ad</p><p>Since d ≠ 0 (non-constant A.P.): a = 8d</p><p><strong>Step 5:</strong> Find the common ratio r of the G.P.:</p><p>r = a₅/a₂ = (a + 4d)/(a + d) = (8d + 4d)/(8d + d) = 12d/9d = 4/3</p><p>∴ Answer: <strong>4/3</strong></p>
Correct Answer: C