<p>The fifth term of a G.P. is 2. Find the product of the first 9 terms of the G.P.</p>
Step-by-Step Solution
Key Concept: In a G.P., the product of terms equidistant from the ends equals the product of the middle term with itself. For 9 terms, the 5th term is the middle term, so the product of all 9 terms equals (5th term)^9.
<p><strong>Step 1:</strong> Let the G.P. be a, ar, ar², ar³, ar⁴, ar⁵, ar⁶, ar⁷, ar⁸.</p><p><strong>Step 2:</strong> Use the property that in a G.P. with odd number of terms, the product equals (middle term)^(number of terms).</p><p><strong>Step 3:</strong> For 9 terms, the middle (5th) term is ar⁴ = 2.</p><p><strong>Step 4:</strong> Product of all 9 terms = (ar⁴)⁹ = (2)⁹ = 512.</p><p><strong>Verification:</strong> Product = a·ar·ar²·ar³·ar⁴·ar⁵·ar⁶·ar⁷·ar⁸ = a⁹r³⁶ = (ar⁴)⁹ = 2⁹ = 512.</p><p>∴ Answer: <strong>512</strong></p>
Correct Answer: B