the 1s and the 2s alternate

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What real number is equal to the expression $2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1 + \cdots}}}$ , where the 1s and the 2s alternate?

Guest Nov 23, 2020

Alan · Answer 1 · 2020-11-23T11:30:25

Treat this as follows: $x = 2+\frac{1}{1+\frac{1}{x}}$

or $x =2+\frac{x}{1+x}$

Rearrange this to obtain a quadratic equation in x and solve for x (only the positive solution will apply).