MathematicsThe value of continued fraction 1+11+11+⋯ is

The value of continued fraction $1 + \frac{1}{1 + \frac{1}{1 + \dots}}$ is

A
$\frac{\sqrt{5} + 1}{2}$
B
$\frac{\sqrt{5} - 1}{2}$
C
$\frac{\sqrt{5} + 1}{3}$
D
$\frac{\sqrt{5} - 1}{3}$

Solution:

$1 + \frac{1}{1 + \frac{1}{1 + \dots}} = x$

$1 + \frac{1}{1 + \frac{1}{1 + \dots}} = x$

$1 + \frac{1}{x} = x$

$x + 1 = x^{2}$

$x^{2} - x - 1 = 0$

$x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 + \sqrt{5}}{2}$

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