Using
R3205: Probability mass function for geometric random positive integer, we have for some geometric random number $M$ with parameter $\theta$
\begin{equation}
\mathbb{P}(M = n)
= \mathbb{P}(N - 1 = n)
= \mathbb{P}(N = n + 1)
= \theta (1 - \theta)^n
\end{equation}
$\square$
