Find the value of: \(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)
+26389 Find the value of:
\(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\) ?
\(\begin{array}{|rcll|} \hline x &=& 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}} \\ x-1 &=& \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}} \\ x-1 &=& \cfrac{1}{2 + \cfrac{1}{2 + \ddots}} \\ && \boxed{x-1= \cfrac{1}{2 + \ddots} } \\ x-1 &=& \cfrac{1}{2 + (x-1) } \\ x-1 &=& \cfrac{1}{x+1} \\ (x-1)(x+1) &=& 1 \\ x^2-1 &=& 1 \\ x^2 &=& 2 \\ \mathbf{x} &=& \mathbf{\sqrt{2}} \\ \hline \end{array}\)
