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# HELP PLEASE I DON'T UNDERSTAND

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

May 7, 2019

#1
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Add 1 to both sides     and we have that

1

x + 1  =    2  +       __________

2 +  1

________

2 +  .......

So we have that

x  =   1 +    1

______          multiply  through by   x + 1

x + 1

x(x + 1)  = (x + 1)  + 1

x^2 + 1x =  1x + 2         subtract 1x from both sides

x^2    =  2          take the positive square root.....since the original right side is positive

x = √2   May 7, 2019
edited by CPhill  May 7, 2019
#2
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$$\text{let }a = \dfrac{1}{2+\dfrac{1}{2+\dfrac{1}{2+\ddots}}}$$

$$a = \dfrac{1}{2+a}\\ a^2+2a-1=0\\ a = \dfrac{-2\pm \sqrt{4+4}}{2} = -1 \pm \sqrt{2}\\ \text{we can rule out the negative solution since everything is positive}\\ a = -1 + \sqrt{2}$$

$$x= 1+a = \sqrt{2}$$

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May 7, 2019
#3
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Find the value of

$$x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.$$

$$\begin{array}{|rcll|} \hline \mathbf{x} &=& \mathbf{1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}} } \\\\ x-1 &=& \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}} \\\\ \dfrac{1}{x-1} &=& 2+ \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}} \\\\ \dfrac{1}{x-1} &=& 2+ (x-1) \\\\ 1 &=& (x-1)(2+x-1) \\ 1 &=& (x-1)(x+1) \\ 1 &=& x^2-1 \\ x^2 &=& 2 \\ \mathbf{x} &=& \mathbf{\sqrt{2}} \\ \hline \end{array}$$ May 7, 2019