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Find the value of $\[x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.\]$

 

I was struggling with this problem for a bit, could someone help? Thank you!

 May 22, 2019
 #1
avatar+102355 
+1

\(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}. \)

 

Add 1 to both sides

 

x + 1  =   2 +        1

                    ____________

                                 1

                     2  +  ________      

                             2 +   ....                 (1)

 

So....going back to the orirginal equation and substituting (1) we get

 

 

x  =   1   +      1

                  _____           mutiply both sides by x + 1

                    x + 1

 

 

 

x(x + 1)  =  (x + 1) + 1

 

x^2 + x  =  x + 2     subtract x from both sides

 

x^2  =  2         take both roots

 

x =  ± √2

 

But.....since the right side of the original  is positive....then we need the positive root

 

So   

 

x  =   √2

 

cool cool cool

 May 22, 2019
 #2
avatar+22884 
+2
 May 23, 2019

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