$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = \frac{1+\sqrt{5}}{2} = \phi$, is this a coincidence? I was playing around with square roots today when I "discovered" this.  
$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$
$\sqrt{1 + x} = x$
$1 + x = x^2$ 
Which, via the quadratic formula, leads me to the golden ratio.  
Is there any significance to this or is it just a random coincidence?  

Let $a_n$ include $n$'s 1. Then $a_{n+1}=\sqrt{1+a_n}$. One can show by induction that $a_n$ is increasing and is bounded from above by $3$. Induction steps:
$$
a_{n+1}=\sqrt{1+a_n}\geq\sqrt{1+a_{n-1}}=a_n; a_{n+1}=\sqrt{1+a_n}<\sqrt{1+3}=2<3.
$$
So there exists $L=\lim a_n$ satisfying $L=\sqrt{1+L}$, which implies $L$ equals the value you find.
