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Compute \(\cfrac{9}{9 + \cfrac{9}{9 + \cfrac{9}{9 + \dotsb}}}\)

 May 25, 2020
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Call this "x" : \(\cfrac{9}{9 + \cfrac{9}{9 + \cfrac{9}{9 + \dotsb}}} \)

Which simplifies to: \(\frac{9}{9+x}\)

 

From this, we can set up the equations:
\(x=\frac{9}{9+x}\)

\(x^2+9x=9\)

\(x^2+9x-9=0\)---> This quadratic equation had 2 solutions, one positive and one negative, for obvious reasons we can rule out the negative solution. This means that the positive solution is our answer:

\(x=\frac{-9+3\sqrt{13}}{2}\)

I appreciate your time!


*Note: I'm not 100% sure about my answer. If anyone sees anything wrong with my thinking process, make sure to point it out!

 May 25, 2020

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