Question

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

Answer 1

hint : use this substitution
x-1=y

so \[y=2+\frac{1}{2+\frac{1}{2+}...}\]

Answer 2

How does that get me anywhere....?

Answer 3

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

Answer 4

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

Answer 5

oh

Answer 6

because the entire bottom part is same as "y" :

\[y=2+\frac{1}{\color{red}{2+\frac{1}{2+}...}} = 2+\frac{1}{\color{red}{y}}\]

Answer 7

Oh... I understand now. Thank you

Answer 8

you should get \(y = \sqrt{2}-1\)

thus \(x = y+1 = \sqrt{2}\)

Answer 9

\[y=2+\frac{1}{y}\\y^2=2y+1\\y^2-2y-1=0\\(y-1)^2-2=0\\(y-1)=\pm \sqrt{2}\\y=1+\sqrt{2}\]because y>0