Find the value of $\[x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.\]$
I was struggling with this problem for a bit, could someone help? Thank you!
\(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}. \)
Add 1 to both sides
x + 1 = 2 + 1
____________
1
2 + ________
2 + .... (1)
So....going back to the orirginal equation and substituting (1) we get
x = 1 + 1
_____ mutiply both sides by x + 1
x + 1
x(x + 1) = (x + 1) + 1
x^2 + x = x + 2 subtract x from both sides
x^2 = 2 take both roots
x = ± √2
But.....since the right side of the original is positive....then we need the positive root
So
x = √2