<p>Find the value of \(2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2+\cdots\infty}}}\).</p>
Step-by-Step Solution
Key Concept: Recognize this infinite continued fraction as a self-similar expression: if x equals the entire expression, then x = 2 + 1/x, which reduces to a quadratic equation. Solve for x and select the positive root that satisfies the constraint x > 2.
<p><strong>Step 1:</strong> Let x = 2 + 1/(2 + 1/(2 + 1/(2 + ⋯))). Since the pattern repeats infinitely, the denominator is also equal to x.</p><p><strong>Step 2:</strong> Write the self-similar equation: x = 2 + 1/x</p><p><strong>Step 3:</strong> Multiply both sides by x: x² = 2x + 1</p><p><strong>Step 4:</strong> Rearrange to standard form: x² − 2x − 1 = 0</p><p><strong>Step 5:</strong> Apply the quadratic formula: x = (2 ± √(4 + 4))/2 = (2 ± √8)/2 = (2 ± 2√2)/2 = 1 ± √2</p><p><strong>Step 6:</strong> Since the continued fraction begins with 2 and all terms are positive, x must be greater than 2. Check: 1 + √2 ≈ 2.414 > 2 ✓ and 1 − √2 ≈ −0.414 < 0 ✗</p><p>∴ Answer: 1 + √2</p>
Correct Answer: 1 + \sqrt{2}