SL 1.3—Geometric sequences and series

You already know that arithmetic sequences increase or decrease by a fixed common difference $d$. Geometric sequences work differently , instead of adding the same value each time, you multiply by a fixed value called the common ratio.

Geometric Sequence: A sequence in which each term is obtained by multiplying the previous term by a fixed constant called the common ratio, $r$. The sequence is defined by its first term $u_1$ and the value of $r$.

The general form of a geometric sequence looks like this:
$u_1,u_{1}r,u_1{r}^2,u_1{r}^3,\dots ,u_1{r}^{n-1}$

Notice that the power of $r$ is always one less than the position of the term , the 1st term has $r^0=1$, the 2nd has $r^1$, the 3rd has $r^2$, and so on.