Let $N \in \text{Geometric}(\theta)$ be a D4001: Geometric random positive integer.
Let $n \in 1, 2, 3, \ldots$ be a D5094: Positive integer.

Let $X_1, X_2, X_3, \ldots \in \mathsf{Bernoulli}(\theta)$ be an independent collection of Bernoulli random numbers such that \begin{equation} N \overset{d}{=} \min \left\{ K \in 0, 1, 2, \ldots : \sum_{k = 1}^K X_k = 1 \right\} \end{equation} We have \begin{equation} \{ N = n \} = \{ X_1 = 0, X_2 = 0, \ldots, X_{n - 1} = 0, X_n = 1 \} \end{equation} Thus, by independence \begin{equation} \begin{split} \mathbb{P} (N = n) & = \mathbb{P} (X_1 = 0, X_2 = 0, \ldots, X_{n - 1} = 0, X_n = 1) \\ & = \left( \prod_{m = 1}^{n - 1} \mathbb{P}(X_m = 0) \right) \mathbb{P}(X_n = 1) \\ & = (1 - \theta)^{n - 1} \theta \end{split} \end{equation} $\square$