Sum of Geometric Sequence

Theorem
Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$.

Let $n \in \N_{>0}$.

Then:
 * $\displaystyle \sum_{j \mathop = 0}^{n - 1} x^j = \frac {x^n - 1} {x - 1}$

Also presented as
Note that when $x < 1$ the result is usually given as:
 * $\displaystyle \sum_{j \mathop = 0}^{n - 1} x^j = \frac {1 - x^n} {1 - x}$

Some sources give it as:
 * $\displaystyle \sum_{j \mathop = 0}^n x^j = \frac {1 - x^{n + 1} } {1 - x}$

and likewise its corollary:
 * $\displaystyle \sum_{j \mathop = 0}^n a x^j = a \paren {\frac {1 - x^{n + 1} } {1 - x} }$

Also see

 * Sum of Infinite Geometric Sequence