;; -*- mode: imath -*-
;; Please type C-c $ to convert all maxima and latex forms into images.
;; (C) 2004 Yasuaki Honda (yhonda@mac.com)

Q: Find the value of a formula {latex \sqrt{\, n + \sqrt{n + \sqrt{n + \cdots}}} latex} for {latex n \in N latex}.

A: Let's define x as the value of the given formula:
{latex x= \sqrt{\, n + \sqrt{n + \sqrt{n + \cdots}}} latex}. 
Then x satisfies the formula {maxima x=sqrt(n+x) maxima}&{latex  x=\isqrt{x+n} latex}. 
Remember {maxima x>0 maxima}&{latex  x>0 latex}, take square of the formula, we obtain 
{maxima x^2=x+n maxima}&{latex 
 x^{2}=x+n latex}. The solution of the formula is:

{maxima solve(x^2=x+n, x) maxima}&{latex  \left[ x=-\ifracn{\isqrt{4\*n+1}-1}{2},\linebreak[0]x=\ifracn{\isqrt{4\*n+1}+1}{2} \right]  latex}

Since {maxima x>0 maxima}&{latex  x>0 latex}, we know that the latter is the answer.
{maxima second(solve(x^2=x+n, x)) maxima}&{latex  x=\ifracn{\isqrt{4\*n+1}+1}{2} latex}.