<p>Fifth term of a G.P. is 2, then the product of its 9 terms is</p>
Step-by-Step Solution
Key Concept: In a G.P., the product of terms equidistant from the ends equals the product of middle terms. For 9 terms, the middle (5th) term is the geometric mean of all terms, 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 (9 terms total).</p><p><strong>Step 2:</strong> The 5th term is ar⁴ = 2 (given).</p><p><strong>Step 3:</strong> Product of all 9 terms = a · ar · ar² · ar³ · ar⁴ · ar⁵ · ar⁶ · ar⁷ · ar⁸ = a⁹r^(0+1+2+3+4+5+6+7+8) = a⁹r^36 = (ar⁴)⁹ = 2⁹ = 512.</p><p><strong>Alternative insight:</strong> In a G.P. with odd number of terms, Product = (middle term)^(number of terms) = (ar⁴)⁹ = 2⁹ = 512.</p><p>∴ Answer: <strong>512</strong></p>
Correct Answer: B