Common Ratio
The common ratio is the constant multiplier between consecutive terms in a geometric sequence.
Formula
r = a_{n+1} / a_n
Definition
The common ratio is the number you multiply each term by to get the next term in a geometric sequence, the same value between every pair of consecutive terms: $r = a_{n+1} / a_n$ for all $n$. It determines the sequence's behavior: $r > 1$ gives growth, $0 < r < 1$ gives decay toward $0$, $r = -1$ alternates between two values, and $|r| < 1$ is required for an infinite geometric series to converge. The common ratio is an element of the multiplicative group $(\mathbb{R}\setminus\{0\}, \times)$ or $(\mathbb{C}\setminus\{0\}, \times)$, and the convergence condition depends on the chosen metric: in non-Archimedean (p-adic) metrics, $|r|_p < 1$ can hold for rationals that are large in the usual sense.
Example
In $5, 15, 45, 135, \ldots$, the common ratio is $3$, and in $80, 40, 20, 10, \ldots$, it is $1/2$. A radioactive substance losing $20\%$ of its mass each year has $r = 0.80$: starting at $100$g gives $100, 80, 64, 51.2, 40.96, \ldots$, with the ratio $0.80$ encoding the decay rate. In p-adic analysis, the geometric series $\sum_{n=0}^{\infty} p^n$ converges to $1/(1-p)$ in the p-adic metric since $|p|_p = 1/p < 1$, even though $p > 1$ as a real number.
Key Insight
You can find the common ratio by dividing any term by the term before it; if the answer is the same every time, the sequence is geometric, and $r$ is the base of the exponential function underlying the sequence, so knowing $r$ and one term completely determines it. The convergence condition $|r| < 1$ is deeply tied to the chosen metric: different metrics on the same field produce different notions of what "small" means, changing which geometric series converge.