A sequence in which the ratio between consecutive terms is constant (each term is found by multiplying the previous term by the same number).
A sequence is an ordered list of numbers (called terms) such as $6, 42, 294, \ldots$.
In many practical settings, the terms change by the same multiplier each step, rather than by the same amount.
That is exactly the idea of a geometric sequence.
A sequence $u_n$ is geometric if there is a constant number $r$ such that $$\frac{u_{n+1}}{u_n}=r$$ for every term where this division is defined.
Note
Different books use different letters.
You may see the first term written as $a$ and the ratio as $q$.
We use $u_1$ for the first term and $r$ for the common ratio.
Understand the Common Ratio and How It Controls Behavior
Definition
Common ratio
The constant multiplier $r$ in a geometric sequence, found by dividing a term by the previous term: $r=\frac{u_{n+1}}{u_n}$.
Example
If a geometric sequence begins $4, 12, 36, 108, \ldots$, then $$r=\frac{12}{4}=\frac{36}{12}=\frac{108}{36}=3,$$ so every term is the previous term multiplied by 3.
Different Values of $r$ Produce Different Patterns
The value of the common ratio $r$ determines the long-term behavior.
$r>1$: if $u_1>0$, the terms increase rapidly (geometric growth).
$0<r<1$: if $u_1>0$, the terms decrease toward 0 (geometric decay).
$r=1$: all terms are equal (constant sequence).
$r<0$: the terms alternate in sign, because multiplying by a negative number flips the sign each step.
Example
The sequence $5, -10, 20, -40, 80, \ldots$ is geometric with $$r=\frac{-10}{5}=-2$$
Because $r$ is negative, the signs alternate: $+, -, +, -, +, \ldots$.
Note
When $r$ is negative, the terms of a geometric sequence alternate in sign.
The sizes still grow if $|r|>1$.
Common Mistake
The constant sequence $3, 3, 3, 3, \ldots$ is geometric (with $r=1$).
However, the all-zero sequence $0, 0, 0, 0, \ldots$ is usually not treated as geometric because ratios like $\frac{0}{0}$ are undefined, so the "constant ratio" test breaks down.
Use Recursive and Explicit Formulae Correctly
Two types of rules are used to describe sequences.
Recursive Form for a Geometric Sequence
If the first term is $u_1$ and the common ratio is $r$, the recursive definition is $$u_{n+1}=r u_n,\quad u_1\text{ given}$$
This matches the idea "multiply by $r$ to get the next term."
Explicit Form for a Geometric Sequence
Starting from $u_1$:
$u_2=u_1r$
$u_3=u_1r^2$
$u_4=u_1r^3$
So the general (explicit) term is $$u_n=u_1r^{n-1}$$
Hint
To check the power $n-1$, substitute $n=1$: you should get $u_1$.
Indeed $u_1r^0=u_1$.
Find Missing Information by Forming Equations
Many questions give partial information (for example, two terms, or a relationship between terms) and ask you to determine $u_1$, $r$, or a later term.
The explicit formula is especially powerful because it turns patterns into algebra.
For $u_n=6\cdot 7^{n-1}$, find the first term to exceed one million.
Solve the inequality: $$6\cdot 7^{n-1}>1\,000\,000$$
Divide by 6: $$7^{n-1}>\frac{1\,000\,000}{6}\approx 166\,666.7$$
Take logarithms: $$(n-1)\log 7>\log(166\,666.7)$$
So $$n-1>\frac{\log(166\,666.7)}{\log 7}\approx 6.18,$$ hence $n>7.18$.
The first integer value that works is $n=8$.
Therefore, the first term to exceed one million is $$u_8=6\cdot 7^{7}=4\,941\,258$$
Exam technique
In inequalities like $u_1r^{n-1}>k$, isolate the exponential part first, then use logs, then round $n$ up to the next integer (because $n$ is a term number).
Decide Whether a Sequence Is Geometric (or Not)
To test whether a sequence is geometric, check whether the ratios of consecutive terms are constant.
A reliable method:
Compute $\frac{u_2}{u_1}$ and $\frac{u_3}{u_2}$.
If these ratios are equal (and division is defined), the sequence is geometric.
Example
Let's classify each of the following sequences:
$18, 9, 4.5, 2.25, \ldots$
$\frac{9}{18}=\frac{4.5}{9}=\frac{2.25}{4.5}=0.5$
So it is geometric with $r=0.5$.
$ 1, 4, 9, 16, 25, \ldots$
Ratios are $4, \frac{9}{4}, \frac{16}{9}, \ldots$ (not constant), so it is not geometric.
$13, 39, 117, 351, \ldots$
$\frac{39}{13}=\frac{117}{39}=\frac{351}{117}=3$
So it is geometric with $r=3$.
Common Mistake
Do not confuse "constant difference" (arithmetic) with "constant ratio" (geometric).
For $4, 7, 10, 13, \ldots$ the differences are constant (3), but the ratios are not.
Link Geometric Sequences to Real Contexts
Geometric sequences model situations where the same percentage change happens repeatedly.
Compound interest: multiply by $(1+i)$ each time period.
Population growth at a fixed rate: multiply by $(1+g)$ each period.
Depreciation/decay at a fixed percentage: multiply by $(1-d)$ each period.
Case study
An investment of \$150 at 5% compound interest per year follows a geometric sequence with $u_1=150$ and $r=1.05$.
After 3 years, you have applied the multiplier $1.05$ three times: $$u_4=150\cdot 1.05^{3}\approx 173.64$$
Here term 1 is the initial amount (year 0), then terms 2, 3, 4 are after 1, 2, 3 years.
Active recall
A sequence has $u_1=12$ and $u_3=48$. Find the possible values of $r$.
For $u_1=5$ and $r=-\frac{1}{2}$, write the first four terms.
Is $0, 0, 0, 0, \ldots$ geometric? Explain using the ratio idea.