AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsSequences & Exponential FunctionsGeometric Sequences

Geometric Sequences (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Geometric sequences

What is a geometric sequence?

A geometric sequence is a sequence in which successive terms have a common ratio
- This means each term is obtained by multiplying the previous term by the same constant
- This common ratio represents a constant proportional change
- The common ratio is usually denoted by $r$
E.g. the sequence $3, 6, 12, 24, 48, \dots$ is geometric with a common ratio of $r = 2$
- Each term is 2 times the previous term
  - $3 \times 2 = 6$
  - $6 \times 2 = 12$
  - $12 \times 2 = 24$
  - etc.
The common ratio can be any nonzero value:
- $r > 1$ : the terms grow away from zero (for positive-valued sequences)
- $0 < r < 1$ : the terms shrink toward zero (for positive-valued sequences)
- $r < 0$ : the terms alternate in sign

How can I find the common ratio of a geometric sequence?

To find the common ratio of a geometric sequence
- Divide any term by the term before it
  - $r = \frac{g_{n + 1}}{g_{n}}$
If the ratios between consecutive terms are not all equal, then the sequence is not geometric

What is the general term of a geometric sequence?

The general term (also called the nth term) of a geometric sequence can be written in two forms
Using the initial value
- $g_{n} = g_{0} r^{n}$
  - where $g_{0}$ is the initial value (the term when $n = 0$ )
  - and $r$ is the common ratio
Using any known term
- $g_{n} = g_{k} r^{(n - k)}$
  - where $g_{k}$ is the value of the $k$ th term
  - and $r$ is the common ratio
- This form is useful when you don't know $g_{0}$ but you do know a different term
Both forms express the same idea
- Start from a known term and multiply by the common ratio the appropriate number of times

How do these formulas work in practice?

E.g. a geometric sequence has initial value $g_{0} = 5$ and a common ratio of $r = 3$
- The general term is $g_{n} = 5 \cdot 3^{n}$
- So $g_{0} = 5$ , $g_{1} = 5 \times 3^{1} = 15$ , $g_{2} = 5 \times 3^{2} = 45$ , $g_{3} = 5 \times 3^{3} = 135$ , $\dots$
Or e.g. you are told that $g_{2} = 48$ and $r = \frac{1}{2}$
- Using the second form: $g_{n} = 48 \cdot {(\frac{1}{2})}^{(n - 2)}$
- You can verify that to check the answer
  - $g_{2} = 48 \cdot {(\frac{1}{2})}^{2 - 2} = 48 \cdot {(\frac{1}{2})}^{0} = 48 \cdot 1 = 48$ ✓
- And also find other terms
  - E.g. the initial term is $g_{0} = 48 \cdot {(\frac{1}{2})}^{0 - 2} = 48 \cdot {(\frac{1}{2})}^{- 2} = 48 \cdot 4 = 192$

How does a geometric sequence grow compared to an arithmetic sequence?

An increasing arithmetic sequence increases equally with each step
- the same amount is added each time
An increasing geometric sequence (with positive values) increases by a larger amount with each successive step
- because the same ratio is applied to an ever-larger value
E.g. consider the geometric sequence $2, 6, 18, 54, 162, \dots$ (ratio $r = 3$ )
- The increases between terms are $4, 12, 36, 108, \dots$
  - getting larger each time
Compare this to the arithmetic sequence $2, 6, 10, 14, 18, \dots$ (difference $d = 4$ )
- The increases between terms are always $4$
This distinction between additive change (arithmetic) versus multiplicative change (geometric) is a key idea that carries over into the study of linear and exponential functions

What does the graph of a geometric sequence look like?

Like all sequences, the graph of a geometric sequence consists of discrete points at whole number values of $n$
For a geometric sequence with $r > 1$ (and positive initial value), the points curve upward with increasing steepness
For a geometric sequence with $0 < r < 1$ (and positive initial value), the points curve downward, getting closer and closer to zero

Two graphs comparing geometric sequences: left graph increases with r>1, right graph decreases with 0<r<1, both plotted against n. — **Graphs of increasing (r>1) and decreasing (0<r<1) geometric sequences**

Worked Example

Graph on a grid with horizontal axis labeled n and vertical axis labeled gₙ. Points are plotted at (1,9), (3,3) and (4,1), and then also at n=5, n=6 and n=7 for values of gₙ that continue to decrease.

Values of the terms of a geometric sequence $g_{n}$ are graphed in the figure. Which of the following is an expression for the $n$ th term of the geometric sequence?

(A) $g_{n} = 3 {(\frac{1}{3})}^{(n - 3)}$

(B) $g_{n} = 9 {(3)}^{(n - 2)}$

(D) $g_{n} = 27 {(\frac{1}{3})}^{n}$

Answer:

Consider the values of $g_{n}$ for the first three points on the graph

I.e. for $n = 2, n = 3$ and $n = 4$

$9, 3, 1, . . .$

They are going down by a factor of $\frac{1}{3}$ each time

I.e., $9 \times \frac{1}{3} = 3$ and $3 \times \frac{1}{3} = 1$

That means that the common ratio is $\frac{1}{3}$

That rules out option (B), which has a common ratio of 3
Don't be fooled by the fact that when $n = 2$ , $g_{2} = 9 {(3)}^{(2 - 2)} = 9 \cdot 3^{0} = 9 \cdot 1 = 9$
- That point agrees with the graph
- But that formula will not give correct values for $n = 3, 4, . . .$

Test out the values of the other three options when $n = 2$

option (A): $g_{2} = 3 {(\frac{1}{3})}^{(2 - 3)} = 3 {(\frac{1}{3})}^{- 1} = 3 \cdot 3 = 9$

option (C): $g_{2} = 9 {(\frac{1}{3})}^{(2 - 3)} = 9 {(\frac{1}{3})}^{- 1} = 9 \cdot 3 = 27$

option (D): $g_{2} = 27 {(\frac{1}{3})}^{2} = 27 \cdot \frac{1}{9} = 3$

Only option (A) gives the correct value for $g_{2}$

(A) $g_{n} = 3 {(\frac{1}{3})}^{(n - 3)}$

Examiner Tips and Tricks

If you remember the general form for a sequence

$g_{n} = g_{k} r^{(n - k)}$

then you should be able to spot right away that option (A) in the above Worked Example gives the correct form for the sequence in the graph with $k = 3$ , where $r = \frac{1}{3}$ and $g_{3} = 3$ .

Worked Example

Values of the terms of a geometric sequence $g_{n}$ are given in the table below.

$n$	0	1	2	3	4
$g_{n}$	6	18	54	162	486

(a) Find the common ratio $r$ of the sequence.

Answer:

The common ratio is found by dividing consecutive terms:

$r = \frac{g_{1}}{g_{0}} = \frac{18}{6} = 3$

This can be verified by checking with other terms

$\frac{g_{2}}{g_{1}} = \frac{54}{18} = 3$ , $\frac{g_{3}}{g_{2}} = \frac{162}{54} = 3$ , etc.

$r = 3$

(b) Write an expression for the general term $g_{n}$ .

Answer:

Use the formula $g_{n} = g_{0} r^{n}$

with $g_{0} = 6$ and $r = 3$ :

$g_{n} = 6 \cdot 3^{n}$

Answer:

Substitute $n = 7$ into the equation from part (b)

$g_{7} = 6 \cdot 3^{7} = 6 \cdot 2187 = 13, 122$

$g_{7} = 13, 122$

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent

Geometric Sequences (College Board AP® Precalculus): Revision Note

Geometric sequences

What is a geometric sequence?

How can I find the common ratio of a geometric sequence?

What is the general term of a geometric sequence?

How do these formulas work in practice?

How does a geometric sequence grow compared to an arithmetic sequence?

What does the graph of a geometric sequence look like?

Worked Example

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis