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What real number is equal to the expression \($2 + \frac{4}{1 + \frac{4}{2 + \frac{4}{1 + \cdots}}}$\), where the $1$s and the $2$s alternate?

 Oct 12, 2017
 #1
avatar+100588 
+1

Let   y  =    2  +          4

                            _______

                           1    +      4

                                       _____

                                        ........

 

So   we have

 

y   =    2       +       4

                          _____

                          1  +   4

                                  __

                                    y

 

 

y  =  2   +       4

                   _____

                 [  y  + 4 ]  / y

 

 

 

y =  2    +  4y /  [ y + 4]        multiply through by  y + 4

 

y [ y + 4]  = 2 [ y + 4]   +  4y     simplify

 

y^2  + 4y  =  2y + 8  +  4y

 

y^2  - 2y  - 8    = 0       factor

 

(y - 4)  ( y + 2)  = 0

 

Set each factor to 0  and solve for y  and we have that

 

y = 4    or  y  = -2

 

The continued fraction is positive....so ...y  = 4  =  the real number

 

P.S.  - as I'm never too sure about these, could someone check my answer  ???

 

 

cool cool cool

 Oct 12, 2017
edited by CPhill  Oct 12, 2017
 #3
avatar+598 
+1

This is correct. Thanks!

michaelcai  Oct 12, 2017
 #2
avatar
0


 

 Oct 12, 2017
edited by Guest  Oct 12, 2017

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