Common Ratio
Functions & Advanced AlgebraThe 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. It is the same (common) between every pair of consecutive terms.
Example
In $5, 15, 45, 135, \ldots$, the common ratio is $3$ (multiply by $3$ each time). In $80, 40, 20, 10, \ldots$, the common ratio is $1/2$.
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, you have a geometric sequence.
Definition
In a geometric sequence, the common ratio $r = a_{n+1} / a_n$ for all $n$. It determines the behavior: $r > 1$ (growth), $0 < r < 1$ (decay toward $0$), $r = -1$ (alternates between two values), $|r| < 1$ required for infinite series to converge.
Example
A radioactive substance loses $20\%$ of its mass each year: $r = 0.80$. Starting at $100$g: $100, 80, 64, 51.2, 40.96, \ldots$ The common ratio $0.80$ encodes the decay rate.
Key Insight
The common ratio is the base of the exponential function underlying the sequence. Knowing $r$ and one term completely determines the entire sequence.
Definition
The common ratio $r$ is an element of the multiplicative group $(\mathbb{R}\setminus\{0\}, \times)$ or $(\mathbb{C}\setminus\{0\}, \times)$. For the infinite geometric series $\sum a r^n$ to converge, we need $|r| < 1$ (in the standard metric). In non-Archimedean metrics (like p-adic), convergence criteria differ: $|r|_p < 1$ may hold for rationals that are large in the usual sense.
Example
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
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.