Alright, so I have a question on a little open-book challenge-test thingy that deals with repeating square roots, in a form as follows...
$\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$
Repeated 2012 times (2012 total square roots)
Over and over and over again. I am newish to TeX, so I am not exactly sure how to model it the way it shows up on paper. It looks sorta like:
$s_n = \sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}$
How is something like this simplified?
Working it out logically (I am a highschool freshman, mind you), I get something like this for my example: $3-\frac{1}{6^{2011}}$
Is this correct? It seems like I could use some sort of limit to prove this, but I have not officially gone through anything beyond Geometry. Now, I do own bits and pieces of knowledge when it comes to calculus and such, but not enough to count on with this sorta thing ;)
EDIT|IMPORTANT: This is what I need to prove: $3 > \sqrt{6+\sqrt{6+\sqrt{6+\cdots}}} > 3-\frac{1}{5^{2011}}$
Where the \cdots
means however many more square roots are needed to make a total of 2012