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

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Arithmetic sequences

What is an arithmetic sequence?

An arithmetic sequence is a sequence in which successive terms have a common difference
- This means the same value is added to each term to get the next term
  - This common difference is also described as a constant rate of change
- The common difference is usually denoted by $d$
E.g. the sequence $3, 7, 11, 15, 19, \dots$ is arithmetic
- with a common difference of $d = 4$
- Each term is 4 more than the previous term
  - $7 - 3 = 4$ , $11 - 7 = 4$ , $15 - 11 = 4$ , etc.
The common difference of a sequence can be
- positive (the sequence increases)
- negative (the sequence decreases)
- or zero (all terms are the same)

How can I find the common difference of an arithmetic sequence?

To find the common difference of an arithmetic sequence
- Subtract any term from the term after it
  - $d = a_{n + 1} - a_{n}$
If the differences between consecutive terms are not all equal, then the sequence is not arithmetic

What is the general term of an arithmetic sequence?

The general term (also called the nth term) of an arithmetic sequence can be written in two forms
Using the initial value
- $a_{n} = a_{0} + d n$
  - where $a_{0}$ is the initial value (the term when $n = 0$ )
  - and $d$ is the common difference

Using any known term
- $a_{n} = a_{k} + d (n - k)$
  - where $a_{k}$ is the value of the known $k$ th term
  - and $d$ is the common difference
- This form is useful when you don't know $a_{0}$ but you do know a different term in the sequence

Both forms express the same idea
- Start from a known term and add the common difference the appropriate number of times

How do these formulas work in practice?

E.g. an arithmetic sequence has $a_{0} = 5$ and a common difference of $d = 3$
- The general term is $a_{n} = 5 + 3 n$
- So: $a_{0} = 5$ , $a_{1} = 8$ , $a_{2} = 11$ , $a_{3} = 14$ , $\dots$
Or e.g. you are told that $a_{4} = 20$ and $d = - 2$
- Using the second form: $a_{n} = 20 + (- 2) (n - 4) = 20 - 2 n + 8 = 28 - 2 n$
- You can verify that to check the answer
  - $a_{4} = 28 - 2 (4) = 28 - 8 = 20$ ✓

What does the graph of an arithmetic sequence look like?

Because an arithmetic sequence is a function of the whole numbers
- its graph is a set of discrete points
- as discussed in the Sequence Basics study guide
The points in the graph of an arithmetic sequence lie along a straight line
- This is because the common difference $d$ acts like a constant slope
- The general term $a_{n} = a_{0} + d n$
  - has the same structure as a linear function $y = b + m x$
However, the graph is not a continuous line
- it is only the individual points at whole number values of $n$

Worked Example

Values of the terms of an arithmetic sequence $a_{n}$ are given in the table below.

$n$	0	1	2	3	4	5
$a_{n}$	14	11	8	5	2	-1

(a) Find the common difference $d$ of the sequence.

Answer:

The common difference is found by subtracting consecutive terms:

$d = a_{1} - a_{0} = 11 - 14 = - 3$

You can check this using other pairs of successive terms

$a_{2} - a_{1} = 8 - 11 = - 3$
$a_{3} - a_{2} = 5 - 8 = - 3$
etc.

$d = - 3$

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

Answer:

Use the formula $a_{n} = a_{0} + d n$

with $a_{0} = 14$ and $d = - 3$

$a_{n} = 14 + (- 3) n$

$a_{n} = 14 - 3 n$

Answer:

Substitute $n = 10$ into the formula from part (b)

$a_{10} = 14 - 3 (10) = 14 - 30 = - 16$

$a_{10} = - 16$

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

Arithmetic sequences

What is an arithmetic sequence?

How can I find the common difference of an arithmetic sequence?

What is the general term of an arithmetic sequence?

How do these formulas work in practice?

What does the graph of an arithmetic sequence look like?

Worked Example

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Author: Roger B

Reviewer: Mark Curtis