To solve the equation:

x = (sqrt(5) - 1) / 2 ≈ 0.618

Step-by-Step Explanation:

1. Identify the Structure: Recognize that the continued fraction repeats infinitely, allowing the inner part to be represented by the same variable x.

2. Set Up the Equation: x = 1 / (1 + x)

3. Solve the Quadratic Equation: Multiply both sides by (1 + x): x(1 + x) = 1

   Expand: x + x² = 1

   Rearrange: x² + x - 1 = 0

4. Apply the Quadratic Formula: x = [-1 ± sqrt(1 + 4)] / 2 x = [-1 ± sqrt(5)] / 2

5. Select the Positive Root: x = (sqrt(5) - 1) / 2 ≈ 0.618

Final Answer: The value of the continued fraction is (sqrt(5) - 1)/2, approximately 0.618.