<p>Let <i>a</i>, <i>b</i> and <i>c</i> be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then \(\dfrac{a}{c}\) is equal to __________.</p>
Step-by-Step Solution
Key Concept: Since a, b, c are in G.P., we have b² = ac. Simultaneously, express a, b, c as terms of an A.P. with first term A and common difference D, then use both conditions to find the relationship between a and c.
<p><strong>Step 1:</strong> Express a, b, c as A.P. terms with first term A and common difference D:</p><p>• a = 7th term = A + 6D</p><p>• b = 11th term = A + 10D</p><p>• c = 13th term = A + 12D</p><p><strong>Step 2:</strong> Since a, b, c are consecutive terms of a G.P., use b² = ac:</p><p>(A + 10D)² = (A + 6D)(A + 12D)</p><p><strong>Step 3:</strong> Expand both sides:</p><p>A² + 20AD + 100D² = A² + 12AD + 6AD + 72D²</p><p>A² + 20AD + 100D² = A² + 18AD + 72D²</p><p><strong>Step 4:</strong> Simplify:</p><p>20AD + 100D² = 18AD + 72D²</p><p>2AD + 28D² = 0</p><p>2D(A + 14D) = 0</p><p><strong>Step 5:</strong> Since A.P. is non-constant, D ≠ 0, so A = -14D</p><p><strong>Step 6:</strong> Find a and c:</p><p>• a = A + 6D = -14D + 6D = -8D</p><p>• c = A + 12D = -14D + 12D = -2D</p><p><strong>Step 7:</strong> Calculate the ratio:</p><p>a/c = (-8D)/(-2D) = 4</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4