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

 Aug 1, 2023
 #1
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Note that \(x=1+\frac1{1+x}\), so \(x^2+x=1+x+1 \implies x=\pm \sqrt 2\). However, since \(x\) is obviously positive, the answer is \(\sqrt 2\).

 Aug 1, 2023

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