What is the value of \(2 + \frac{1}{2 + \frac{1}{2 + \ldots \infty}}\)?
(A) \(1 - \sqrt{2}\)
(B) \(\sqrt{2} + 1\)
(C) \(1 \pm \sqrt{2}\)
(D) none of these
Answer (1)
To solve the expression 2 + 2 + 2 + … ∞ 1 1 , we need to understand the nature of this repeating, infinite continued fraction.
Let's denote the full expression as x :
x = 2 + x 1
This is because the structure of the fraction inside the denominator is self-similar, repeating itself infinitely. Our goal is to solve for x .
Start with the expression:
x = 2 + x 1
Multiply both sides by x to eliminate the fraction:
x 2 = 2 x + 1
Rearrange this equation into standard quadratic format:
x 2 − 2 x − 1 = 0
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c to solve for x , where a = 1 , b = − 2 , and c = − 1 :
x = 2 ⋅ 1 − ( − 2 ) ± ( − 2 ) 2 − 4 ⋅ 1 ⋅ ( − 1 )
x = 2 2 ± 4 + 4
x = 2 2 ± 8
x = 2 2 ± 2 2
x = 1 ± 2
Since x , representing the entire expression, must be greater than 2 , we choose the positive root:
x = 1 + 2
Given the options, the value of the expression is B 2 + 1 .
