SOLUTION: Evaluate √(7/3+√(7/9+√(7/3+√(7/9+⋯)) ) )

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Question 1106487: Evaluate √(7/3+√(7/9+√(7/3+√(7/9+⋯)) ) )
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


x = sqrt(7/3+sqrt(7/9+sqrt(7/3+sqrt(7/9+...))))

x^2 = 7/3 + sqrt(7/9+sqrt(7/3+sqrt(7/9)...)))

x^2-7/3 = sqrt(7/9+sqrt(7/3+sqrt(7/9)...)))

(x^2-7/3)^2 = 7/9+sqrt(7/3+sqrt(7/9)...))

(x^2-7/3)^2 = 7/9 + x

x^4-(14/3)x^2+49/9 = 7/9 + x

x^4-(14/3)x^2-x+42/9 = 0

x^4-(14/3)x^2-x+14/3 = 0

3x^4-14x^2-3x+14 = 0

This polynomial has two real roots, x=1 and x=2, and a pair of complex roots.

Clearly the given expression is real, because it is the square root of a positive number. And clearly the expression is not equal to 1. Therefore the value of the expression is 2.
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.
Evaluate
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I just solved similar, but much more simple problem under the link
https://www.algebra.com/algebra/homework/Angles/Angles.faq.question.1106262.html

https://www.algebra.com/algebra/homework/Angles/Angles.faq.question.1106262.html


        So,  I will assume that it was YOU who requested the preceding problem  (because  "Lightning never strikes the same place twice").

        In other words,  I will assume that you are familiar with the idea and the solution of that previous problem.

        Based on it,  I will be short with this one. 

Let  us consider, for brewity of writing, more general expression

 = x,

where  a = ,  b = .  Then  

 = x.                  (It is clear, and I will not spend words to justify it . . . )


It is equivalent to


 -  = x,    or

 = 0.


Now substitute here  a = ,  b = . You will get this equation in the form

 = 0,   or, multiplying all the terms by 3

 = 0.


Now I will not go into details, and simply show the plot of the last polynomial.

It clearly shows that x= 2 is the root.  And now you can check it MANUALLY  (as I did . . . )






Plot y = 



It makes me CONVINCED that   = 2.


Check.   = 1.984 (approx.)

Solved.


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