Sequences as
Functions & Recursion
Discover how every sequence is secretly a function on the integers, and master the art of recursive definitions that build each term from the ones before it.
What Is a Sequence?
A sequence is an ordered list of numbers that follows a definite pattern. Each number in the list is called a term, and every term has a specific position, called its index.
Sequences appear everywhere: population counts year by year, monthly savings deposits, the Fibonacci spiral in nature. Understanding their structure lets us predict future terms and analyze growth.
Sequences Are Functions
A sequence is a function whose domain is a subset of the integers (usually \( \{1, 2, 3, \ldots\} \) or \( \{0, 1, 2, \ldots\} \)) and whose range is the set of term values.
Function Notation for Sequences
Because a sequence is a function, we can write it using function notation. There are two common styles:
Explicit vs Recursive Definitions
There are two ways to define a sequence function. Both produce the same terms, but they work very differently:
Anatomy of a Recursive Definition
Every recursive definition has two essential parts. Miss either one and the sequence is undefined.
The Fibonacci Sequence
The most famous recursive sequence uses two base cases and looks back two terms:
Each term is the sum of the two preceding terms. Because the rule references \( f(n) \) and \( f(n-1) \), we need two base cases to get started.
Explicit vs Recursive: Side by Side
| Explicit | Recursive | |
|---|---|---|
| Formula type | \( a_n = \text{expression in } n \) | \( a_n = \text{expression in } a_{n-1} \) |
| To find \( a_{100} \) | Plug in \( n = 100 \) directly | Must compute all terms 1 through 99 first |
| Base case needed? | No | Yes, always |
| Example | \( a_n = 2n + 1 \) | \( a_1 = 3,\; a_n = a_{n-1} + 2 \) |
| Best for | Direct calculation | Describing a process step by step |
How to Evaluate a Recursive Sequence
Example 1: Arithmetic Sequence
Example 2: Geometric Sequence
Notice the exponential growth: each term doubles. Explicit form: \( g(n) = 3 \cdot 2^{n-1} \). The graph curves upward steeply because multiplication compounds.
Example 3: Computing Fibonacci Terms
Why the Domain Matters
A sequence's domain is a subset of the integers. This is what makes it different from a continuous function like \( f(x) = 2x + 1 \).
Writing a Recursive Definition
Given the sequence 4, 7, 10, 13, 16, ..., write both an explicit and recursive definition.
Common Mistakes to Avoid
Summary: Types of Sequence Definitions
| Arithmetic | Geometric | Fibonacci-type | |
|---|---|---|---|
| Recursive rule | \( a_n = a_{n-1} + d \) | \( a_n = r \cdot a_{n-1} \) | \( a_n = a_{n-1} + a_{n-2} \) |
| Base cases | 1 needed | 1 needed | 2 needed |
| Explicit form | \( a_n = a_1 + d(n-1) \) | \( a_n = a_1 \cdot r^{n-1} \) | No simple closed form |
| Growth pattern | Linear | Exponential | Near-exponential |
Question 1 of 5
A sequence is defined by \( a_1 = 2 \) and \( a_n = a_{n-1} + 5 \). What is \( a_4 \)?
Question 2 of 5
Which statement is true about the domain of a sequence?
Question 3 of 5
A recursive definition is missing its base case: \( a_n = 3 \cdot a_{n-1} \). Why is this definition incomplete?
Question 4 of 5
The Fibonacci sequence is defined by \( f(0) = 1,\; f(1) = 1,\; f(n+1) = f(n) + f(n-1) \) for \( n \geq 1 \). What is \( f(6) \)?
Question 5 of 5
Which of the following is a valid recursive definition for the sequence 6, 18, 54, 162, ...?