jmc

We start out with the equation $n(n+1)=2550$. Expanding, we find $n^2+n-2550=0$. This factors to $(n-50)(n+51)=0$, so $n=50\text{ or }-51.$ Since $n$ must be negative, we have $n=-51$, hence our two integers are $n=-51$ and $n+1=-50$, which add up to $\boxed{-101}$.