Sequences & Series
Geometric Progression
Grade 11
Question:
<p>Given that \(ar^{n+1} = ar^n + ar^{n-1}\), find the value of the common ratio \(r\).</p>
<p>\(r = \dfrac{\sqrt{5}+1}{2}\)</p>
<p>\(r = \dfrac{\sqrt{5}-1}{2}\)</p>
<p>\(r = \dfrac{1+\sqrt{5}}{3}\)</p>
<p>\(r = \sqrt{5}\)</p>
Step-by-Step Solution
Key Concept: Divide the given equation by ar^(n-1) to eliminate the exponential terms and obtain a quadratic equation in r. This transforms the recurrence relation into a characteristic equation.
<p><strong>Step 1:</strong> Start with the given equation: ar^(n+1) = ar^n + ar^(n-1)</p><p><strong>Step 2:</strong> Divide both sides by ar^(n-1) (assuming a ≠ 0, r ≠ 0):</p><p>r² = r + 1</p><p><strong>Step 3:</strong> Rearrange to standard quadratic form:</p><p>r² - r - 1 = 0</p><p><strong>Step 4:</strong> Apply the quadratic formula: r = (1 ± √(1+4))/2 = (1 ± √5)/2</p><p><strong>Step 5:</strong> The two solutions are r = (1+√5)/2 (the golden ratio φ) and r = (1-√5)/2</p><p>∴ Answer: B (typically r = (1+√5)/2 or both roots depending on options)</p>
Correct Answer: B