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# Clever Algebraic Manipulation Needed

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Let $$x = {1+\frac{\sqrt{2}}{1+\frac{\sqrt{2}}{1+ \dotsb}}}.$$

Find $$\frac{1}{(x+1)(x-2)}$$. When your answer is in the form $$\frac{A+\sqrt{B}}{C}$$ , where A , B , and C are integers, and B is not divisible by the square of a prime, what is $$|A|+|B|+|C|$$?

Aug 5, 2022

#2
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This is just

$$x=1+\frac{\sqrt2}{x}$$

Now take it from there.

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Aug 7, 2022