## h0pe one year ago Find the value of $x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.$

1. amoodarya

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

2. h0pe

How does that get me anywhere....?

3. ganeshie8

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

4. h0pe

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

5. h0pe

oh

6. ganeshie8

because the entire bottom part is same as "y" : $y=2+\frac{1}{\color{red}{2+\frac{1}{2+}...}} = 2+\frac{1}{\color{red}{y}}$

7. h0pe

Oh... I understand now. Thank you

8. ganeshie8

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

9. 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