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# help!

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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:

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