SOLUTION: Dear MathTutor, I appreciate your response on the question I asked regarding the repeated fractions, however, I have failed to grasp a certain part of it. Ref: https://www.al

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Question 1199973: Dear MathTutor,
I appreciate your response on the question I asked regarding the repeated fractions, however, I have failed to grasp a certain part of it.
Ref: https://www.algebra.com/tutors/students/your-answer.mpl?question=1199949
Could you please explain why +x+-+1+ can replace that chunk due to its "repeated nature".
I understand the repeated nature, but not how it justifies this substitution. Please elaborate. Really appreciate it!

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!



Anywhere we see "x-1", we can replace it with that massive nested continued fraction on the right hand side.

But buried within the continued fraction, is a copy of itself. It's fractal in nature.

That's how we're able to go from this

to this
x+-+1+=+1%2F%281+%2B+1%2F%282%2Bhighlight%28%28x-1%29%29%29%29

The stuff in the red box can be replaced with x-1 since x-1 is equal to that very stuff.

-----------------------------------

Another example of this process

x = sqrt(2+sqrt(2+sqrt(2+...)))
x = sqrt(2+sqrt(2+sqrt(2+...)))
x = sqrt(2+x)
The stuff in red represents x since x is the repeated nested square root.