Sequences & Series
Geometric Progression
Grade 11
Question:
<p>Let \(a_1, a_2, \ldots, a_{10}\) be a G.P. If \(\dfrac{a_3}{a_1} = 25\), then \(\dfrac{a_9}{a_5}\) equals:</p>
<p>\(4(5^2)\)</p>
<p>\(5^3\)</p>
<p>\(5^4\)</p>
<p>\(2(5^2)\)</p>
Step-by-Step Solution
Key Concept: In a G.P., the ratio between terms depends only on the difference in their positions (indices), not their absolute positions. If a₃/a₁ = r² where r is common ratio, then a₉/a₅ = r⁴ = (r²)².
<p><strong>Step 1:</strong> In a G.P. with first term a and common ratio r: aₙ = ar^(n-1)</p><p><strong>Step 2:</strong> Calculate a₃/a₁:</p><p>a₃/a₁ = (ar²)/(a) = r² = 25</p><p><strong>Step 3:</strong> Calculate a₉/a₅:</p><p>a₉/a₅ = (ar⁸)/(ar⁴) = r⁴ = (r²)² = 25² = 625</p><p>∴ Answer: C (625)</p>
Correct Answer: C