You need to use the starred environments to remove the automatic equation numbering i.e. \begin{equation*
} and \end{equation*}
:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{equation*}
\sum_{1}^{n} =
\left\{\begin{array}{lr}
1, & n=1\\
n+\sum_{1}^{n-1}, &\forall n>1
\end{array}\right.
\end{equation*}
\end{document}
A more succinct way that reduces typing is to follow the advice from @barbara beeton and use \[
and \]
to wrap the one-line equation in:
\documentclass{article}
\usepackage{mathtools}
\begin{document}
\[
\sum_{1}^{n} =
\left\{\begin{array}{lr}
1, & n=1\\
n+\sum_{1}^{n-1}, &\forall n>1
\end{array}\right.
\]
\end{document}
which produces the same output image.