P3395
We can write
\begin{equation}
\begin{split}
q^{N + 1} - 1
= q^{N + 1} - q^0
& = \left( \sum_{n = 1}^{N + 1} q^n \right) - \left( \sum_{n = 0}^N q^n \right) \\
& = q \left( \sum_{n = 0}^N q^n \right) - \left( \sum_{n = 0}^N q^n \right) \\
& = (q - 1) \left( \sum_{n = 0}^N q^n \right)
\end{split}
\end{equation}
Dividing each side by the nonzero quantity $q - 1$, we then get
\begin{equation}
\left( \sum_{n = 0}^N q^n \right)
= \frac{q^{N + 1} - 1}{q - 1}
= \frac{(-1) (1 - q^{N + 1})}{(-1) (1 - q)}
= \frac{1 - q^{N + 1}}{1 - q}
\end{equation}
$\square$
