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# What real number is equal to the expression ​, where the $1$s and the $2$s alternate?

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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?

michaelcai  Oct 12, 2017
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#1
+76972
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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  ???

CPhill  Oct 12, 2017
edited by CPhill  Oct 12, 2017
#3
+281
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This is correct. Thanks!

michaelcai  Oct 12, 2017
#2
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Guest Oct 12, 2017
edited by Guest  Oct 12, 2017

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