Question

Simplify the expression: \( 4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \cdots}}} \)

• A. \( 2 + \sqrt{5} \)
• B. \( 2 - \sqrt{5} \)
• C. \( 2 + \sqrt{3} \)
• D. \( 2 - \sqrt{3} \)

Hint:

For continued fractions, set up an equation where the fraction repeats, solve it using algebraic methods, and apply the quadratic formula for solutions.

Answer 1

The Correct Option is B

Let \( x = 4 + \frac{1}{4 + \frac{1}{4 + \cdots}} \). This is a continued fraction, and we can express it as: \( x = 4 + \frac{1}{x} \) Multiplying both sides by \( x \): \( x^2 = 4x + 1 \) Now, subtract \( 4x + 1 \) from both sides: \( x^2 - 4x - 1 = 0 \) This is a quadratic equation. Using the quadratic formula: \( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-1)}}{2(1)} \) \( x = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2} \) \( x = \frac{4 \pm 2\sqrt{5}}{2} \) \( x = 2 \pm \sqrt{5} \) Since \( x \) must be positive, we take the positive root: \( x = 2 + \sqrt{5} \) Thus, the value of the expression is \( 2 + \sqrt{5} \), corresponding to option (2).