A geometric sequence is one where each term is multiplied by a fixed number called the common ratio $r$.
$$r = \frac{\text{any term}}{\text{previous term}}$$
nth term formula: $$u_n = ar^{n-1}$$
where $a$ is the first term and $r$ is the common ratio.
Example: The sequence $3, 6, 12, 24, \ldots$ has $a = 3$ and $r = 2$. $$u_5 = 3 \times 2^{4} = 48$$
Finding $r$: divide any term by the one before it. If the sequence is decreasing, $0 < r < 1$. If terms alternate in sign, $r$ is negative.
Identifying a geometric sequence: check that the ratio between consecutive terms is constant — unlike an arithmetic sequence where the difference is constant.
Common error: confusing geometric sequences with arithmetic ones — always check by dividing, not subtracting.