Question 1106262
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<pre>
Let x = {{{sqrt(5 + sqrt(5 + sqrt(5 + ellipsis)))}}}.     (1)


It is what you ask to simplify.


Square both sides of (1). You will get

x^2 = 5 + x,     or

x^2 - x - 5 = 0.


Use the quadratic formula:

{{{x[1,2]}}} = {{{(1 +- sqrt(1^2 + 4*5))/2}}} = {{{(1 +- sqrt(21))/2}}}.


Since x, obviously, must be > 0, only positive root is the solution:  x = {{{(1 + sqrt(21))/2}}}.


<U>Answer</U>.  {{{sqrt(5 + sqrt(5 + sqrt(5 + ellipsis)))}}} = {{{(1 + sqrt(21))/2}}}.
</pre>

Solved !



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Amazingly, isn't it ?



Let me show you another, even more amazing calculation !



Simplfy  {{{sqrt(2 + sqrt(2 + sqrt(2 + ellipsis)))}}}.


<B>Solution</B>


<pre>
Let x = {{{sqrt(2 + sqrt(2 + sqrt(2 + ellipsis)))}}}.     (2)


It is what I want to simplify.


Square both sides of (2). You will get

x^2 = 2 + x,     or

x^2 - x - 2 = 0.


Use the quadratic formula:

{{{x[1,2]}}} = {{{(1 +- sqrt(1^2 + 4*2))/2}}} = {{{(1 +- sqrt(9))/2}}} = {{{(1 +- 3)/2}}}.


Since x, obviously, must be > 0, only positive root is the solution:  x = {{{(1 + 3)/2}}} = 2.


<U>Answer</U>.  {{{sqrt(2 + sqrt(2 + sqrt(2 + ellipsis)))}}} = 2.
</pre>


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