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Geometric Sequences (College Board AP® Precalculus): Study Guide

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)

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

$g_{7} = 13, 122$

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Geometric Sequences (College Board AP® Precalculus): Study Guide

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