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Sequence Basics (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Sequence basics

What is a sequence?

A sequence is a function from the whole numbers to the real numbers
- The whole numbers are $0, 1, 2, 3, 4, \dots$
- Each whole number input is mapped to a real number output
  - The outputs are known as the terms of the sequence
  - The inputs are known as the term numbers

The terms of a sequence are typically written as $a_{0}, a_{1}, a_{2}, a_{3}, \dots$
- or sometimes starting from $a_{1}$ instead of from $a_{0}$
$a_{n}$ is the nth term of the sequence
- I.e. it is the output value corresponding to the input value (term number) $n$
  - So $a_{0}$ is the output corresponding to $n = 0$
    - $a_{1}$ is the output corresponding to $n = 1$
    - $a_{2}$ is the output corresponding to $n = 2$
    - etc.
The nth term can be expressed as a formula in terms of $n$
- E.g. $a_{n} = 2 n + 3$
  - Then $a_{0} = 2 (0) + 3 = 3$
    - $a_{1} = 2 (1) + 3 = 5$
    - $a_{2} = 2 (2) + 3 = 7$
    - etc.
Because a sequence is a function, each input value (term number) corresponds to exactly one output value

How is the graph of a sequence different from the graph of a typical function?

The domain of a sequence is the whole numbers
- This is a discrete set
  - i.e. isolated, individual values rather than a continuous range
This means the graph of a sequence consists of discrete points instead of a continuous curve
- You plot individual dots, not a connected line or curve
Compare this to functions you've seen previously (like polynomials or rational functions), whose domain typically includes all real numbers or large intervals of real numbers

Two graphs: the left shows a cubic polynomial curve; the right displays discrete points of a related sequence, each with labelled axes. — **Graph of a function (connected curve) versus the graph of the equivalent sequence (discrete points)**

Why are sequences important in this course?

In AP® Precalculus, sequences are used primarily as a stepping stone to help you understand other types of functions
- Arithmetic sequences (which have a constant difference between successive terms) are closely connected to linear functions
- Geometric sequences (which have a constant ratio between successive terms) are closely connected to exponential functions
Studying how quantities change step-by-step in a sequence helps build intuition for how related functions behave

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