$\frac{1}{1+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{9}}+...+\frac{1}{\sqrt{2001}+\sqrt{2005}}+\frac{1}{\sqrt{2005}+\sqrt{2009}}$
$=\frac{1-\sqrt{5}}{(1+\sqrt{5})(1-\sqrt{5})}+\frac{\sqrt{5}-\sqrt{9}}{(\sqrt{5}+\sqrt{9})(\sqrt{5}-\sqrt{9})}+...+\frac{\sqrt{2001}-\sqrt{2005}}{(\sqrt{2001}+\sqrt{2005})(\sqrt{2001}-\sqrt{2005})}+\frac{\sqrt{2005}-\sqrt{2009}}{(\sqrt{2005}+\sqrt{2009})(\sqrt{2005}-\sqrt{2009})}$
$=\frac{1-\sqrt{5}}{1-5}+\frac{\sqrt{5}-\sqrt{9}}{5-9}+...+\frac{\sqrt{2001}-\sqrt{2005}}{2001-2005}+\frac{\sqrt{2005}-\sqrt{2009}}{2005-2009}$
$=\frac{1-\sqrt{5}}{-4}+\frac{\sqrt{5}-\sqrt{9}}{-4}+...+\frac{\sqrt{2001}-\sqrt{2005}}{-4}+\frac{\sqrt{2005}-\sqrt{2009}}{-4}$
$=\frac{1-\sqrt{5}+\sqrt{5}-\sqrt{9}+...+\sqrt{2001}-\sqrt{2005}+\sqrt{2005}-\sqrt{2009}}{-4}$
$=\frac{1-\sqrt{2009}}{-4}$
$=\frac{\sqrt{2009}-1}{4}$
