Question 1106487
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Evaluate {{{sqrt(7/3 + sqrt(7/9 + sqrt(7/3 + sqrt(7/9) + ellipsis)))}}}
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I just solved similar, but much more simple problem under the link

<A HREF=https://www.algebra.com/algebra/homework/Angles/Angles.faq.question.1106262.html>https://www.algebra.com/algebra/homework/Angles/Angles.faq.question.1106262.html</A>


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



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


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


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


<pre>
Let  us consider, for brewity of writing, more general expression

{{{sqrt(a + sqrt(b + sqrt(a + sqrt(b) + ellipsis)))}}} = x,

where  a = {{{7/3}}},  b = {{{7/9}}}.  Then  

{{{(x^2-a)^2 - b}}} = x.                  (It is clear, and I will not spend words to justify it . . . )


It is equivalent to


{{{x^4 -2a*x^2 + a^2}}} - {{{b}}} = x,    or

{{{X^4 - 2a*x^2 - x + (a^2-b)}}} = 0.


Now substitute here  a = {{{7/3}}},  b = {{{7/9}}}. You will get this equation in the form

{{{x^4 - (14/3)*x^2 - x + 14/3}}} = 0,   or, multiplying all the terms by 3

{{{3x^4 -14x^2 - 3x + 14}}} = 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 . . . )



{{{graph( 330, 330, -1.5, 5.5, -10.5, 15.5,
          3x^4 - 14x^2 - 3x + 14
)}}}


Plot y = {{{3x^4 - 14x^2 - 3x + 14}}}



It makes me CONVINCED that  {{{sqrt(7/3 + sqrt(7/9 + sqrt(7/3 + sqrt(7/9) + ellipsis)))}}} = 2.


<U>Check</U>.  {{{sqrt(7/3 + sqrt(7/9 + sqrt(7/3 + sqrt(7/9))))}}} = 1.984 (approx.)
</pre>

Solved.