<p><strong>23.</strong> Let \(\{t_n\}\) be a sequence of integers in G.P. in which \(t_4 : t_6 = 1:4\) and \(t_2 + t_5 = 216\). Then \(t_1\) is</p>
Step-by-Step Solution
Key Concept: In a G.P., use the ratio condition t₄:t₆ = 1:4 to find the common ratio r, then apply the sum condition t₂ + t₅ = 216 to determine t₁. Since terms must be integers, verify that the common ratio satisfies this constraint.
<p><strong>Step 1:</strong> Use the ratio condition. In a G.P., tₙ = t₁·rⁿ⁻¹.</p><p>Given t₄:t₆ = 1:4, we have:</p><p>$$\frac{t_4}{t_6} = \frac{t_1 r^3}{t_1 r^5} = \frac{1}{r^2} = \frac{1}{4}$$</p><p>Therefore r² = 4, so <strong>r = ±2</strong></p><p><strong>Step 2:</strong> Use t₂ + t₅ = 216:</p><p>$$t_1 r + t_1 r^4 = 216$$</p><p>$$t_1(r + r^4) = 216$$</p><p><strong>Case 1: r = 2</strong></p><p>$$t_1(2 + 16) = 216$$</p><p>$$t_1 · 18 = 216$$</p><p>$$t_1 = 12$$ ✓ (integer)</p><p><strong>Case 2: r = -2</strong></p><p>$$t_1(-2 + 16) = 216$$</p><p>$$t_1 · 14 = 216$$</p><p>$$t_1 = \frac{216}{14} = \frac{108}{7}$$ ✗ (not an integer)</p><p><strong>Step 3:</strong> Since {tₙ} is a sequence of integers, only r = 2 works.</p><p>∴ Answer: <strong>t₁ = 12</strong> (Option A)</p>
Correct Answer: A