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

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Question 1205842: Evaluate sqrt(5 + sqrt(5^2 + sqrt(5^4 + sqrt(5^8 + ...))))
Answer by ikleyn(52810) About Me  (Show Source):
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Evaluate sqrt(5 + sqrt(5^2 + sqrt(5^4 + sqrt(5^8 + ...)))).
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Let x = sqrt%285+%2B+sqrt%285%5E2+%2B+sqrt%285%5E4+%2B+sqrt%285%5E8+%2B+ellipsis%29%29%29%29.


Square both sides

    x%5E2 = 5+%2B+sqrt%285%5E2+%2B+sqrt%285%5E4+%2B+sqrt%285%5E8+%2B+ellipsis%29%29%29%29.


Move 5 from right side to the left

    x%5E2+-+5 = sqrt%285%5E2+%2B+sqrt%285%5E4+%2B+sqrt%285%5E8+%2B+ellipsis%29%29%29%29.


Transform right side

    x%5E2+-+5 = sqrt%285%29+%2A+sqrt%285+%2B+sqrt%285%5E2+%2B+sqrt%285%5E4+%2B+ellipsis%29%29%29%29.


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

    x%5E2+-+5 = %28sqrt%285%29%29%2Ax,  or

    x%5E2 - sqrt%285%29%2Ax - 5 = 0.


Find the root x using the quadratic formula

    x%5B1%2C2%5D = %28sqrt%285%29+%2B-+sqrt%28%28sqrt%285%29%29%5E2-4%2A%28-5%29%29%29%2F2 = %28sqrt%285%29+%2B-+sqrt%2825%29%29%2F2 = %28sqrt%285%29+%2B-+5%29%2F2.


Keep the positive root and reject the negative one.


In this way, you get the ANSWER:  x = %285%2Bsqrt%285%29%29%2F2 = 3.618033989  (rounded).


CHECK.  The partial expression  sqrt%285+%2B+sqrt%285%5E2+%2B+sqrt%285%5E4+%2B+sqrt%285%5E8%29%29%29%29  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.