<p>Let <em>a</em> ∈ (0, 1] satisfies the equation \(a^{2008} - 2a + 1 = 0\) and \(S = 1 + a + a^2 + \ldots + a^{2007}\). The sum of all possible value(s) of \(S\) is</p>
Step-by-Step Solution
Key Concept: From the constraint a²⁰⁰⁸ - 2a + 1 = 0, we get a²⁰⁰⁸ = 2a - 1. Recognize that S is a geometric series sum: S = (1 - a²⁰⁰⁸)/(1 - a) for a ≠ 1, and substitute the constraint to find S.
<p><strong>Step 1:</strong> From the constraint equation a²⁰⁰⁸ - 2a + 1 = 0, we have:</p><p>a²⁰⁰⁸ = 2a - 1</p><p><strong>Step 2:</strong> Check if a = 1 works: 1 - 2 + 1 = 0 ✓. If a = 1, then S = 1 + 1 + ... + 1 (2008 terms) = 2008.</p><p><strong>Step 3:</strong> For a ∈ (0, 1), use the geometric series formula:</p><p>S = (1 - a²⁰⁰⁸)/(1 - a)</p><p><strong>Step 4:</strong> Substitute a²⁰⁰⁸ = 2a - 1:</p><p>S = (1 - (2a - 1))/(1 - a) = (2 - 2a)/(1 - a) = 2(1 - a)/(1 - a) = 2</p><p><strong>Step 5:</strong> For a ∈ (0, 1) with a ≠ 1, we get S = 2. Combined with a = 1 giving S = 2008, the two possible values are S = 2 and S = 2008.</p><p><strong>Step 6:</strong> Sum of all possible values = 2 + 2008 = 2010</p><p>∴ Answer: B</p>
Correct Answer: B