Sequences & Series
AP and GP
Grade 11
Question:
<p>Given <em>a</em>, <em>b</em> and <em>c</em> be the 7th, 11th and 13th terms respectively of a non-constant A.P. Let <em>A</em> be the first term and <em>D</em> be the common difference of A.P. If <em>a</em>, <em>b</em>, <em>c</em> are in G.P., find \(\dfrac{a}{c}\).</p>
Step-by-Step Solution
Key Concept: Express a, b, c in terms of A and D using the A.P. formula, then apply the G.P. condition (b² = ac) to create an equation in A and D. The non-constant condition ensures D ≠ 0, allowing you to solve for the ratio a/c.
<p><strong>Step 1:</strong> Express terms using A.P. formula where A is first term, D is common difference.</p><p>a = A + 6D (7th term)</p><p>b = A + 10D (11th term)</p><p>c = A + 12D (13th term)</p><p><strong>Step 2:</strong> Apply G.P. condition: b² = ac</p><p>(A + 10D)² = (A + 6D)(A + 12D)</p><p>A² + 20AD + 100D² = A² + 12AD + 6AD + 72D²</p><p>A² + 20AD + 100D² = A² + 18AD + 72D²</p><p><strong>Step 3:</strong> Simplify the equation</p><p>20AD + 100D² = 18AD + 72D²</p><p>2AD + 28D² = 0</p><p>2D(A + 14D) = 0</p><p><strong>Step 4:</strong> Since A.P. is non-constant, D ≠ 0</p><p>Therefore: A + 14D = 0, so A = -14D</p><p><strong>Step 5:</strong> Calculate a/c</p><p>a = A + 6D = -14D + 6D = -8D</p><p>c = A + 12D = -14D + 12D = -2D</p><p>∴ a/c = (-8D)/(-2D) = 4</p>
Correct Answer: 4