A geometric sequence multiplies by the same number each time: $3, 6, 12, 24, \ldots$ multiplies by $2$. That number is the common ratio $r$. It is the multiplying cousin of an arithmetic sequence, and, like the exponential functions you just compared with lines, it grows by equal ratios.
Why it works
To reach the $n$th term from the first, you multiply by $r$ once for each step, and there are $n - 1$ steps (from term $1$ to term $2$ is one step). So $a_n = a_1 \cdot r^{n-1}$. This is an exponential function of the position $n$: it equals $\frac{a_1}{r} \cdot r^n$.
Key idea
$a_n = a_1 \cdot r^{n-1}$, where $r = \frac{a_{n+1}}{a_n}$ is the same for every pair of neighbors.
Worked examples
Worked example: A far term
Find the $8$th term of $5, 15, 45, \ldots$
- $a_1 = 5$ and $r = \frac{15}{5} = 3$.
- $a_8 = 5 \cdot 3^{7} = 5 \cdot 2187 = 10935$.
Answer: $10935$
Worked example: Halving
Write a formula for $64, 32, 16, \ldots$ and find $a_7$.
- $a_1 = 64$ and $r = \frac{1}{2}$, so $a_n = 64\left(\frac{1}{2}\right)^{n-1}$.
- $a_7 = 64 \cdot \frac{1}{64} = 1$.
Answer: $a_n = 64\left(\frac{1}{2}\right)^{n-1}$; $a_7 = 1$.
Worked example: Alternating signs
What is the ratio of $2, -6, 18, -54, \ldots$?
- $\frac{-6}{2} = -3$, and $\frac{18}{-6} = -3$.
Answer: $r = -3$: a negative ratio makes the signs alternate.
Common mistake
Using $n$ instead of $n - 1$ in the exponent, or adding the ratio instead of multiplying. Check any formula at $n = 1$: $a_1 \cdot r^{0} = a_1$ gives back the first term, but $a_1 \cdot r^1$ would give the second. And $3, 6, 12$ is not “add $3$”: $12 - 6 = 6$, but $\frac{12}{6} = \frac{6}{3} = 2$.