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$x = 1 + \cfrac{\sqrt{3}}{1 + \cfrac{\sqrt{3}}{1 + \dotsb}}$.  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 $|A| + |B| + |C|$?

 Jun 15, 2021
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First, simplify the first equation:

\(x=1+\frac{\sqrt{3}}{x}\\x^2-x=\sqrt{3}\)

Then, notice that \(\frac{1}{(x+1)(x-2)}\) simplifies to \(\frac{1}{x^2-x-2}\), so we can replace it like so:

\(\frac{1}{\sqrt{3}-2}\)

To simplify, just multiply the top and bottom by the conjugate

\(\frac{\sqrt{3}+2}{(\sqrt{3}+2)(\sqrt{3}-2)}\\=\frac{\sqrt{3}+2}{3-4}=\frac{2+\sqrt{3}}{-1}\)

\(|2|+|3|+|-1|=\boxed{6}\)

 Jun 15, 2021

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