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Context: I was given an expression to evaluate which is as follows:-

$$r = \sqrt{\frac{20}{\sqrt{\frac{20}{\sqrt{\frac{20}{\sqrt{\frac{20}{\dots}}}}}}}}$$

I am writing this expression as follows:

$$ r = \sqrt{\frac{20}{r}}$$

Am I right in doing so? I am not satisfied with my approach.

Question: Is there any other way to approach this problem or what I am doing is right and I am overthinking it

Bill Dubuque
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Rohit
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  • Welcome to MSE. Is there a reason why the "obvious" answer of $r^3=20^2=400$ should not work? – Red Five Jun 14 '24 at 07:51
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    Red Five $r^3 = 20$ or $r^3 = 20^2$ ? – Rohit Jun 14 '24 at 07:55
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    Yes, that is an error on my part. Should be $r^3=20$. I was too busy thinking about the general case in an earlier, very similar looking question. – Red Five Jun 14 '24 at 07:57
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    You first have to define the meaning of your infinitely nested radical. Related: https://math.stackexchange.com/questions/3119631 Take an arbitrary (positive!) value for $x_0$ and define a sequence by $x_{n+1}=\sqrt{20/x_n}$, and prove that it converges. This will justify your result. – Anne Bauval Jun 14 '24 at 08:02
  • Yes, your approach is correct. +1 – GSmith Jun 14 '24 at 08:17
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    "Yes", your approach can be made correct (see previous comment), and there are tons of analogous posts on this site, where "analogous" means: define $r=f(f(f(\dots)))$, and check that $r=f(r)$. I think it would be better to close as a "duplicate" (I don't know which target to choose) than to post a specific answer. – Anne Bauval Jun 14 '24 at 10:00

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