<p>If the sum of \(n\) terms of a G.P. is \(3 - \dfrac{3^{n+1}}{4^{2n}}\), then find the common ratio.</p>
Step-by-Step Solution
Key Concept: Extract the first term and second term from the given sum formula using S₁ and S₂ - S₁, then use the relationship r = a₂/a₁ to find the common ratio. The sum formula itself encodes both the first term and common ratio.
<p><strong>Step 1:</strong> Given S_n = 3 - 3^(n+1)/4^(2n) = 3 - 3·3^n/16^n</p><p><strong>Step 2:</strong> Find first term: S₁ = 3 - 3·3/16 = 3 - 9/16 = 39/16</p><p>So a = 39/16</p><p><strong>Step 3:</strong> Find second term using S₂: S₂ = 3 - 3·3²/16² = 3 - 27/256 = 741/256</p><p>Therefore a₂ = S₂ - S₁ = 741/256 - 39/16 = 741/256 - 624/256 = 117/256</p><p><strong>Step 4:</strong> Common ratio r = a₂/a₁ = (117/256)/(39/16) = (117/256) × (16/39) = (117×16)/(256×39) = 1872/9984 = 3/16</p><p>∴ Answer: <strong>3/16</strong></p>
Correct Answer: 3/16