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

Question

amoodarya · Answer

hint : use this substitution 
x-1=y

so $$y=2+\frac{1}{2+\frac{1}{2+}...}$$

h0pe · Answer

How does that get me anywhere....?

ganeshie8 · Answer

after that, you just need to solve $$\large y = 2+\frac{1}{y}$$

h0pe · Answer

so it would get me to $$y=\sqrt{2}$$

h0pe · Answer

oh

ganeshie8 · Answer

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 · Answer

Oh... I understand now. Thank you

ganeshie8 · Answer

you should get $y = \sqrt{2}-1$

thus $x = y+1 = \sqrt{2}$

amoodarya · Answer

$$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