Question

The value of 3 + 1/(4 + 1/(3 + 1/(4 + 1/(3 + ⋯ ∞))) is equal to :

• A. 1.5 + √3
• B. 2 + √3
• C. 3 + 2√3
• D. 4 + √3

Hint:

For infinite continued fractions, identify the repeating unit and substitute the whole expression value back into itself.

Answer 1

The Correct Option is A

Step 1: Let $x = 3 + \frac{1}{4 + \frac{1}{x}}$.
Step 2: $x = 3 + \frac{x}{4x + 1} = \frac{12x + 3 + x}{4x + 1}$.
Step 3: $x(4x + 1) = 13x + 3 \implies 4x^2 + x = 13x + 3$.
Step 4: $4x^2 - 12x - 3 = 0$.
Step 5: $x = \frac{12 \pm \sqrt{144 - 4(4)(-3)}}{2(4)} = \frac{12 \pm \sqrt{144 + 48}}{8} = \frac{12 \pm \sqrt{192}}{8}$.
Step 6: $x = \frac{12 + 8\sqrt{3}}{8} = 1.5 + \sqrt{3}$.