[tex]\displaystyle\\
\sqrt{2009 + (1+2+3+ \cdots +2008)\cdot 2}=\\\\\\
=\sqrt{2009 + \frac{2008(2008+1)}{2}\cdot 2}=\\\\\\
=\sqrt{2009 + 2008 \cdot 2009}=\\\\\\
=\sqrt{2009(1 + 2008)} =\sqrt{2009 \cdot 2009} =\sqrt{2009^2} =\boxed{2009 \in N}\\\\\\[/tex]
