<p>Let \(a_n\) be an infinite geometric sequence with a convergent and negative sum. The common ratio of the sequence is \(r\) and the first term is \(a_1\), then which one of the following is always true?</p>
Step-by-Step Solution
Key Concept: For a geometric series to converge with negative sum, we need |r| < 1 (convergence condition) AND a₁ < 0 (negative first term). The sum formula S = a₁/(1-r) is negative only when a₁ and (1-r) have opposite signs.
<p><strong>Step 1:</strong> For convergence of a geometric series: |r| < 1, which means -1 < r < 1.</p><p><strong>Step 2:</strong> The sum is S = a₁/(1-r). Since -1 < r < 1, we have 1-r > 0 always.</p><p><strong>Step 3:</strong> For S < 0 (negative sum) and (1-r) > 0, we need a₁ < 0 (first term must be negative).</p><p><strong>Step 4:</strong> Therefore, the conditions that are always true are:</p><ul><li>|r| < 1 (equivalently: -1 < r < 1)</li><li>a₁ < 0</li><li>The sign of r can be positive or negative (both are possible)</li></ul><p><strong>Step 5:</strong> The statement that is always true must involve a₁ < 0 combined with |r| < 1, or equivalently a₁ < 0 and 0 < 1-r.</p><p>∴ Answer: A</p>
Correct Answer: A