h0pe
  • h0pe
Find the value of \[x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.\]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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amoodarya
  • amoodarya
hint : use this substitution x-1=y so \[y=2+\frac{1}{2+\frac{1}{2+}...}\]
h0pe
  • h0pe
How does that get me anywhere....?
ganeshie8
  • ganeshie8
after that, you just need to solve \[\large y = 2+\frac{1}{y}\]

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h0pe
  • h0pe
so it would get me to \[y=\sqrt{2}\]
h0pe
  • h0pe
oh
ganeshie8
  • ganeshie8
because the entire bottom part is same as "y" : \[y=2+\frac{1}{\color{red}{2+\frac{1}{2+}...}} = 2+\frac{1}{\color{red}{y}}\]
h0pe
  • h0pe
Oh... I understand now. Thank you
ganeshie8
  • ganeshie8
you should get \(y = \sqrt{2}-1\) thus \(x = y+1 = \sqrt{2}\)
amoodarya
  • amoodarya
\[y=2+\frac{1}{y}\\y^2=2y+1\\y^2-2y-1=0\\(y-1)^2-2=0\\(y-1)=\pm \sqrt{2}\\y=1+\sqrt{2}\]because y>0

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