The value of the continued fraction 1+11+11+⋯ is

x=1+11+11+⋯x=1+1xx=x+1xx2−x−1=0x=−(−1)±(−1)2−4×1×−12×1=1±1+42x=1±52x=1+52(Since 'x' must be pos

The value of the continued fraction 1+11+11+⋯ is

x=1+11+11+⋯x=1+1xx=x+1xx2−x−1=0x=−(−1)±(−1)2−4×1×−12×1=1±1+42x=1±52x=1+52(Since 'x' must be pos

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

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$\frac{\sqrt{5} - 1}{2}$

$\frac{\sqrt{5} + 1}{2}$

$\frac{\sqrt{5} - 1}{3}$

$\frac{\sqrt{5} + 1}{3}$

answer is A.

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Detailed Solution

$\begin{array}{l} x = 1 + \frac{1}{1 + \frac{1}{1 + \dots}} \\ x = 1 + \frac{1}{x} \\ x = \frac{x + 1}{x} \\ x^{2} - x - 1 = 0 \\ x = \frac{- (- 1) \pm \sqrt{{(- 1)}^{2} - 4 \times 1 \times - 1}}{2 \times 1} \\ = \frac{1 \pm \sqrt{1 + 4}}{2} \\ x = \frac{1 \pm \sqrt{5}}{2} \\ x = \frac{1 + \sqrt{5}}{2} (Since 'x' must be pos \end{array}$