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Find \(3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3 + ...}}}}\)

 Oct 28, 2019
 #1
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Let the given expression  = x

 

So....we have that

 

x  =  3  +   1 / x

 

x - 3  =  1  / x              multiply through by  x

 

x^2  - 3x   =  1             complete the square on x

 

x^2 - 3x + 9/ 4  =  1 + 9/4

 

(x - 3/2)^2  =  13/4       take the positive root

 

x - 3/2  = √13 / 2

 

x =  [ 3  + √13 ]

      __________

              2

 

 

cool cool cool

 Oct 28, 2019
edited by CPhill  Oct 28, 2019

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