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Find the value of

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

WhoaThere  May 22, 2017
 #1
avatar
+1

Your "continued fraction" adds up to =Sqrt(2) =1.4142135623730.......etc.

Guest May 22, 2017
 #2
avatar+87564 
+2

 

Here's how this is determined :

 

Evaluating from the "bottom" to the "top"  .... we have....

 

2 + 1/2  =  5/2

 

2 + 2/5 = 12/5

 

2 + 5/12  = 29/12

 

2 + 12/29  = 70/29

 

1 + 29/70  =  1.4142857142857143   ...further expansion would get us closer and closer to the √2

 

 

cool cool cool

CPhill  May 22, 2017
 #3
avatar+17743 
+2

Another way:

First:  add 1 to both sides:  x + 1  =  2 + [ 1 / [ 2 + [ 1 / [ 2 + 1 / [ ... ] 

Let  y  =  x + 1,  then:                y  =  2 + [ 1 / [ 2 + [ 1 / [ 2 + 1 / [ ... ] 

But   [ 2 + [ 1 / [ 2 + 1 / [ ... ]  =  y

So:                                            y  =  2 + 1 / y

Multiply by y:                            y2  =  2y + 1

Set equal to 0:             y2 - 2y - 1  =  0

Solve (using the quadratic formula):     y  =  1 +/- sqrt(2)

So:                                                  x + 1  =   1 +/- sqrt(2)

The negative answer can't be correct, so:     x + 1  =  1 + sqrt(2)     --->     x  =  sqrt(2)

geno3141  May 22, 2017
 #4
avatar+87564 
0

Thanks, geno....!!!

 

 

cool cool cool

CPhill  May 22, 2017

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