<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: Express A.P. terms using first term and common difference, then use the G.P. condition that the middle term squared equals the product of outer terms to find the common ratio.
<p><strong>Step 1:</strong> Let the A.P. have first term 'a' and common difference 'd' (d ≠ 0).</p><p>Then: a₂ = a + d, a₅ = a + 4d, a₉ = a + 8d</p><p><strong>Step 2:</strong> Since a₂, a₅, a₉ are in G.P., we have:</p><p>(a₅)² = a₂ · a₉</p><p>(a + 4d)² = (a + d)(a + 8d)</p><p><strong>Step 3:</strong> Expanding the left side:</p><p>a² + 8ad + 16d² = a² + 8ad + ad + 8d²</p><p>a² + 8ad + 16d² = a² + 9ad + 8d²</p><p><strong>Step 4:</strong> Simplifying:</p><p>16d² - 8d² = 9ad - 8ad</p><p>8d² = ad</p><p>8d = a (since d ≠ 0)</p><p><strong>Step 5:</strong> Substituting a = 8d back:</p><p>a₂ = 8d + d = 9d</p><p>a₅ = 8d + 4d = 12d</p><p>a₉ = 8d + 8d = 16d</p><p><strong>Step 6:</strong> Common ratio of G.P.:</p><p>r = a₅/a₂ = 12d/9d = 4/3</p><p>∴ Answer: A (Common ratio = 4/3)</p>
Correct Answer: A