The usual thing to do is replace ( with \left( and ) with \right), which automatically expand to fit the material between them. Note that every \left... requires a \right... (but the type of bracket may be different, i.e. \left(...\right] also works).
I would typeset your equation as
\begin{equation*}
\sum_{i=1}^n i = \left(\sum_{i=1}^{n-1} i\right) + n =
\frac{(n-1)(n)}{2} + n = \frac{n(n+1)}{2}
\end{equation*}
For manual control of sizes (most of the time you won't need these)
For your particular equation, using \left\{ and \right\} to get larger curly braces won't do you much good, as the material being encased by the braces isn't much taller than a parenthesis. Hence, I suggest you use \bigl\{ and \bigr\} for the inner set of braces (the stuff following \min), and \Bigl\{ and \Bigr\} for the outer set:
Incidentally, the use $$ ... $$ to generate display-math equations is heavily deprecated. It's much safer to use \[ and \] (as I do in the example above). For much more on this subject, please see Why is \[ ... \] preferable to $$?
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