SOLUTION: Evaluate sqrt(5 + sqrt(5^2 + sqrt(5^4 + sqrt(5^8 + ...))))

Question 1205842: Evaluate sqrt(5 + sqrt(5^2 + sqrt(5^4 + sqrt(5^8 + ...))))
Answer by ikleyn(52810) (Show Source): You can put this solution on YOUR website!
.
Evaluate sqrt(5 + sqrt(5^2 + sqrt(5^4 + sqrt(5^8 + ...)))).
~~~~~~~~~~~~~~~~~~~~~~~

Let x = .


Square both sides

     = .


Move 5 from right side to the left

     = .


Transform right side

     = .


Notice that the second factor in right side is x, again.
So, your equation is

     = ,  or

     -  - 5 = 0.


Find the root x using the quadratic formula

     =  =  = .


Keep the positive root and reject the negative one.


In this way, you get the ANSWER:  x =  = 3.618033989  (rounded).


CHECK.  The partial expression    is  approximately  3.573355548.

        The partial expression  sqrt(5 + sqrt(5^2 + sqrt(5^4 + sqrt(5^8 + sqrt(16 + sqrt(5^32 + sqrt(5^64 + sqrt(5^128))))))))  

        is approximately  3.61003987.

Solved.

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I understand that I am helping somebody to solve a problem of a Math Olympiad level.

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