Find the value of \(x= 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)
+289 Please be aware that the Guest who 'answered' your question did miss a lot of steps. And there is no such thing as "theory of continued fractions"???
I would recommend you checking this out. https://web2.0calc.com/questions/deliberately-misguiding-answers#r2
Denote y = x-1
y = 1/(2 + 1/....
1/y = 2 + 1(2 + 1/(2..
1/y - 2 = y
y^2 = -2y + 1
y^2 + 2y - 1 = 0
By Quadratic Formula,
y = -1 +- sqrt(2)
Because y has to be nonnegative, y = -1 + sqrt(2)
y = x - 1
-1 + sqrt(2) = x -1
x = sqrt(2)
+289 I am not quite sure if I understand you.
First off, I put 3 question marks to make it apparent that I wasn't actually 100% sure. Instead of thinking that I am assuming things, you shouldn't assume about things yourself.
2. The link I sent about "misguiding answers" is not a suggestion that Guest is doing it all wrong. He did miss a lot of steps and as you know, there have been anynomous peolpe who have been deliberately sending wrong answers.
3. AIME 2019? Interesting which one, I or II? Both of them absolutely don't require such "continued fractions"? If I am mistaken, again, let me know.
4. There is a difference between a theorem and a theory. A theory is something big, like the "BIg Bang Theory". I am pretty sure not anyone would name such a small thing in math, a theory.
5. So you said something about derivatives? Could you explain this "theorem" to me because it looks like it requires calculus and requires more understanding for the person who asked the question.
