<p>Find the value of <br>\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\]</p>
Step-by-Step Solution
Key Concept: Recognize that powers of i repeat with period 4, so reduce all exponents modulo 4, then factor out common terms from numerator and denominator to simplify the geometric series.
<p><strong>Step 1:</strong> Reduce exponents modulo 4. Since i^4 = 1, we have:</p><ul><li>592 ≡ 0 (mod 4), so i^592 = 1</li><li>590 ≡ 2 (mod 4), so i^590 = -1</li><li>588 ≡ 0 (mod 4), so i^588 = 1</li><li>586 ≡ 2 (mod 4), so i^586 = -1</li><li>584 ≡ 0 (mod 4), so i^584 = 1</li></ul><p><strong>Step 2:</strong> Calculate numerator: 1 + (-1) + 1 + (-1) + 1 = 1</p><p><strong>Step 3:</strong> Reduce exponents in denominator modulo 4:</p><ul><li>582 ≡ 2 (mod 4), so i^582 = -1</li><li>580 ≡ 0 (mod 4), so i^580 = 1</li><li>578 ≡ 2 (mod 4), so i^578 = -1</li><li>576 ≡ 0 (mod 4), so i^576 = 1</li><li>574 ≡ 2 (mod 4), so i^574 = -1</li></ul><p><strong>Step 4:</strong> Calculate denominator: -1 + 1 + (-1) + 1 + (-1) = -1</p><p><strong>Step 5:</strong> Divide: 1/(-1) = -1</p><p><strong>Step 6:</strong> Subtract 1: -1 - 1 = -2</p><p>∴ Answer: <strong>-2</strong></p>
Correct Answer: -2