Geometric Sequence
Functions & Advanced AlgebraA 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 each time to get the next term. That multiplier is called the common ratio.
Example
$2, 6, 18, 54, 162, \ldots$ is geometric because you multiply by $3$ each time. Also: $100, 50, 25, 12.5, \ldots$ is geometric with ratio $1/2$ (halving each time).
Key Insight
Geometric sequences appear in exponential growth and decay. A bacteria colony that doubles each hour follows a geometric sequence with ratio $2$.
Definition
In a geometric sequence, each term is a constant multiple $r$ of the previous: $a_{n+1} = r \cdot a_n$. The explicit formula is $a_n = a_1 \cdot r^{n-1}$. If $|r| > 1$, the sequence grows; if $|r| < 1$, it shrinks toward $0$; if $r < 0$, terms alternate in sign.
Example
$a_1 = 3$, $r = 2$. Sequence: $3, 6, 12, 24, 48, \ldots$ The $8$th term: $a_8 = 3 \cdot 2^7 = 3 \cdot 128 = 384$.
Key Insight
Geometric sequences are exponential functions of $n$: $a_n = a_1 \cdot r^{n-1} = (a_1 / r) \cdot r^n$. The connection to exponential functions explains why they model growth and decay.
Definition
A geometric sequence $\{a_n\} = \{a_1 \cdot r^{n-1}\}$ is an exponential function restricted to $\mathbb{N}$. It corresponds to powers of a fixed element in a group. Geometric sequences appear in p-adic analysis (sequences of powers of $p$), spectral theory (geometric decay of eigenvalues), and wavelet theory.
Example
The sequence of reciprocals of factorials $\{1/n!\}$ is not geometric, but 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 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.