Let \(n\in\mathbb N\) be a natural number and let \(a, x \in F\) be any two elements of a given field \((F, +, \cdot)\) with \(x\neq 1\). Then the general sum of the geometric progression \(a, ax, ax^2,\ldots ax^n\) is
\[S_n:=\sum_{0\le k\le n} ax^k=\frac{a-ax^{n+1}}{1-x},\quad\quad (x\neq 1).\]
Proofs: 1
