HMMT February

Algebra Difficulty 4.8 AIME Find the answer

Find the exact value of $1+\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $x$ be what we are trying to find. $x-1=\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}} \Rightarrow \frac{1}{x-1}-1=\frac{2}{1+\frac{1}{1+\frac{2}{1+\cdots}}} \Rightarrow \frac{2}{\frac{1}{x-1}-1}=x \Rightarrow x^{2}-2=0$ , so $x=\sqrt{2}$ since $x>0$ .

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