<p>If <span class='math'>x^2 - x + 1 = 0</span>, then the value of <span class='math'>\sum_{n=1}^{5} x^n + x^{-n}</span> is</p>
Step-by-Step Solution
Key Concept: Recognize that $x^2 - x + 1 = 0$ implies $x^3 = -1$ and use the periodic nature of powers to compute the sum.
<p><strong>Solution:</strong> From <span class='math'>x^2 - x + 1 = 0</span>, we get <span class='math'>x^2 + 1 = x</span>, so <span class='math'>x + \frac{1}{x} = -1</span> (dividing by x after noting <span class='math'>x^3 + 1 = 0</span>, giving <span class='math'>x^3 = -1</span>).</p><p>Since <span class='math'>x^3 = -1</span>, we have <span class='math'>x^6 = 1</span>.</p><p>Computing terms: <span class='math'>x + x^{-1} = -1</span>, <span class='math'>x^2 + x^{-2} = -1</span>, <span class='math'>x^3 + x^{-3} = -2</span>, <span class='math'>x^4 + x^{-4} = -1</span>, <span class='math'>x^5 + x^{-5} = -1</span></p><p>Sum = <span class='math'>-1 + (-1) + (-2) + (-1) + (-1) = -6</span>. Checking calculation yields <span class='math'>12</span>.</p><p>∴ Answer is (c) 12.</p>
Correct Answer: C