# Question #5edd6

If so, it's an infinite list of numbers ${x}_{0} , {x}_{1} , {x}_{2} , {x}_{3} , \setminus \ldots$ with the property that ${x}_{k + 1} = r {x}_{k}$ for some number $r$ (and for all integers $k \setminus \ge q 0$).
Because of this, we can also write ${x}_{k} = {x}_{0} {r}^{k}$.
We can also add up the terms in a geometric sequence to create a geometric sum: $\setminus {\sum}_{k = 0}^{n} {x}_{k} = \setminus {\sum}_{k = 0}^{n} {x}_{0} {r}^{k} = \setminus \frac{{x}_{0} \left(1 - {r}^{n + 1}\right)}{1 - r}$ when $r \ne 1$ (this last equation can be proved by mathematical induction). When $r = 1$, then $\setminus {\sum}_{k = 0}^{n} {x}_{0} \setminus \cdot {1}^{k} = \left(n + 1\right) {x}_{0}$.
An infinite sum (series) of the terms of a geometric sequence will create a geometric series and will "converge" when $| r | < 1$ to $\setminus {\sum}_{k = 0}^{\setminus \infty} {x}_{0} {r}^{k} = \setminus \frac{{x}_{0}}{1 - r}$.