Question

Question #5edd6

Answer 1

Do you mean a geometric sequence ?

If so, it's an infinite list of numbers `x_{0},x_{1},x_{2},x_{3},\ldots` with the property that `x_{k+1}=r x_{k}` for some number `r` (and for all integers `k\geq 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: `\sum_{k=0}^{n}x_{k}=\sum_{k=0}^{n}x_{0}r^{k}=\frac{x_{0}(1-r^{n+1})}{1-r}` when `r!=1` (this last equation can be proved by mathematical induction). When `r=1`, then `\sum_{k=0}^{n}x_{0}\cdot 1^{k}=(n+1)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 `\sum_{k=0}^{\infty}x_{0}r^{k}=\frac{x_{0}}{1-r}`.