<p>If \(b\) is the first term of an infinite G.P. whose sum is five, then \(b\) lies in the interval</p>
Step-by-Step Solution
Key Concept: For an infinite G.P. with first term b and common ratio r, the sum S = b/(1-r) converges only when |r| < 1, and we must solve b/(1-r) = 5 to find the constraint on b.
<p><strong>Step 1:</strong> For an infinite G.P. with first term b and common ratio r, the sum is S = b/(1-r), provided |r| < 1.</p><p><strong>Step 2:</strong> Given that S = 5, we have: b/(1-r) = 5, so b = 5(1-r), which gives r = 1 - b/5.</p><p><strong>Step 3:</strong> For convergence, we need |r| < 1:<br>|1 - b/5| < 1<br>-1 < 1 - b/5 < 1</p><p><strong>Step 4:</strong> From the left inequality: -1 < 1 - b/5 ⟹ b/5 < 2 ⟹ b < 10<br>From the right inequality: 1 - b/5 < 1 ⟹ -b/5 < 0 ⟹ b > 0</p><p><strong>Step 5:</strong> Therefore, b ∈ (0, 10).</p><p>∴ Answer: C</p>
Correct Answer: C