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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,

 Jan 6, 2022
 #1
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Please put your latex into a latex box (on the ribon) so that it displays properly.

 Jan 6, 2022
 #2
avatar+12 
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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,

 

\(x = 1 + \cfrac{\sqrt{3}}{1 + \cfrac{\sqrt{3}}{1 + \dotsb}}\) When we look at this fraction, we see that the denomenator of the fraction is also equal to x, because it's going on forever. Therefore, we can just change this equation to \(x=1+\sqrt{3}/x\)

 

Now, you can just solve for x and plug it into 1/(x+1)(x-2) and get your answer. Hope this helps!

 

(correct me if i am incorrect)

 Jan 6, 2022

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