Follow the logic of geometric distribution, if we want to study the probability of $r^{\text{th}}$ success in $k^{\text{th}}$ of a series of trials, it must be the case that $(r −1)$ success occur during the first $(k −1)$ trials and the $r^{\text{th}}$ happens on exactly the $k^{\text{th}}$ trial.
If we let $X$ be the sum of independent variables $X1, X2, \dots, Xr$, and if $ X \rightarrow \infty$, the negative binomial can be interpreted as $r$ successes happen one after another, and each of which follows the geometric distribution model.
