Geometric Sequence
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed constant called the common ratio.
Formula
a_n = a_1 \cdot r^{n-1}
Definition
A geometric sequence is a pattern where you multiply by the same number, the common ratio $r$, each time to get the next term: $a_{n+1} = r \cdot a_n$, with explicit formula $a_n = a_1 \cdot r^{n-1}$. If $|r| > 1$ the sequence grows, if $|r| < 1$ it shrinks toward $0$, and if $r < 0$ the terms alternate in sign. A geometric sequence is really an exponential function restricted to $\mathbb{N}$, corresponding to powers of a fixed element in a group, and such sequences appear in p-adic analysis (sequences of powers of $p$), spectral theory (geometric decay of eigenvalues), and wavelet theory.
Example
$2, 6, 18, 54, 162, \ldots$ is geometric with ratio $3$, while $100, 50, 25, 12.5, \ldots$ has ratio $1/2$ (halving each time). With $a_1 = 3$ and $r = 2$: the sequence is $3, 6, 12, 24, 48, \ldots$, and the $8$th term is $a_8 = 3 \cdot 2^7 = 384$. The sequence of powers $\{r^n\}$ for $|r| < 1$ has partial sums converging to $1/(1-r)$, the foundational geometric series result underpinning radius-of-convergence analysis in complex analysis.
Key Insight
Geometric sequences appear in exponential growth and decay: a bacteria colony that doubles each hour follows a geometric sequence with ratio $2$, and indeed $a_n = a_1 \cdot r^{n-1} = (a_1 / r) \cdot r^n$ is an exponential function of $n$, explaining why geometric sequences model growth and decay. Geometric sequences are the multiplicative analogs of arithmetic sequences: taking logarithms of a geometric sequence (with $r > 0$) yields an arithmetic sequence, linking the two families through the log-exp correspondence.