Sequences & Series
Common ratio of GP
Grade 11
Question:
<p>In a GP of positive terms, any term is equal to the sum of the next two terms. Which of the following is the common ratio?</p>
<p>1</p>
<p>\( \dfrac{\sqrt{5}-1}{2} \)</p>
<p>\( \dfrac{1-\sqrt{5}}{2} \)</p>
<p>\( \dfrac{2}{\sqrt{5}} \)</p>
Step-by-Step Solution
Key Concept: If any term equals the sum of the next two terms in a GP, then a = ar + ar², which gives us a quadratic equation in r. Since all terms are positive, we need the positive root satisfying 0 < r < 1.
<p><strong>Step 1:</strong> Let the GP have first term 'a' and common ratio 'r'. Any term = ar^(n-1).</p><p><strong>Step 2:</strong> Given condition: ar^(n-1) = ar^n + ar^(n+1)</p><p><strong>Step 3:</strong> Divide by ar^(n-1): 1 = r + r²</p><p><strong>Step 4:</strong> Rearrange: r² + r - 1 = 0</p><p><strong>Step 5:</strong> Using quadratic formula: r = (-1 ± √5)/2</p><p><strong>Step 6:</strong> Since all terms are positive, r must be positive: r = (-1 + √5)/2 ≈ 0.618</p><p><strong>Step 7:</strong> Verify: 0 < r < 1 ✓ (valid for decreasing GP with positive terms)</p><p>∴ Answer: r = (√5 - 1)/2 (which is B)</p>
Correct Answer: B