<p>If \(5 \cdot 2^8 \cdot 3^{16}\) is one of the terms of a G.P. whose first term is 5 and all its terms are natural numbers then possible common ratio of the G.P. is:</p>
Step-by-Step Solution
Key Concept: For a G.P. with first term a=5 and nth term = 5·2^8·3^16, the common ratio r must be such that 5·r^(n-1) = 5·2^8·3^16, giving r^(n-1) = 2^8·3^16. The common ratio r itself must be a natural number (or rational with specific form) to keep all terms natural, and must divide this expression as a perfect power.
<p><strong>Step 1:</strong> Given first term a = 5, and one term of G.P. is 5·2^8·3^16. If this is the nth term, then:</p><p>ar^(n-1) = 5·2^8·3^16</p><p>5·r^(n-1) = 5·2^8·3^16</p><p>r^(n-1) = 2^8·3^16</p><p><strong>Step 2:</strong> For all terms to be natural numbers, r must be a natural number. So r^(n-1) = 2^8·3^16 means r = 2^a·3^b where a(n-1) = 8 and b(n-1) = 16.</p><p><strong>Step 3:</strong> The divisors of gcd(8,16) = 8 are: 1, 2, 4, 8. So (n-1) ∈ {1, 2, 4, 8}</p><p><strong>Step 4:</strong> Possible values:</p><p>• If n-1 = 1: r = 2^8·3^16 ✓</p><p>• If n-1 = 2: r = 2^4·3^8 ✓</p><p>• If n-1 = 4: r = 2^2·3^4 = 4·81 = 324 ✓</p><p>• If n-1 = 8: r = 2^1·3^2 = 2·9 = 18 ✓</p><p><strong>Step 5:</strong> Matching with options A, B, D (these are the composite ratios 2^4·3^8, 2^2·3^4, and 18 respectively)</p><p>∴ Answer: A, B, D</p>
Correct Answer: A,B,D