Sequences as Functions and Recursion

SECTION 1

Historical Context & Motivation

People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain lists of numbers — like 1, 1, 2, 3, 5, 8, 13 — seemed to pop up everywhere in nature, from the spiral of a sunflower to the branching of a tree. These ordered lists are called sequences, and understanding them has been a central goal of mathematics across many cultures and centuries.

~200 BCE

Indian Mathematicians Study Patterns

Scholars like Pingala in ancient India studied rhythmic patterns in poetry and discovered number sequences that would later be known as the Fibonacci numbers.

1202

Fibonacci's Liber Abaci

Leonardo of Pisa, known as Fibonacci, published his famous book introducing the sequence 1, 1, 2, 3, 5, 8… to European mathematics through a rabbit population problem.

1600s

Sequences Meet Algebra

Mathematicians like Pierre de Fermat and Blaise Pascal began writing general formulas and rules for sequences, connecting number patterns to the growing field of algebra.

1800s

Functions Formalized

Mathematicians defined the modern concept of a function — a rule that assigns exactly one output to each input. This allowed sequences to be described as a special type of function whose inputs are integers.

2010

Common Core Standard F-IF.3

The Common Core State Standards formally asked students to recognize that sequences are functions, sometimes defined recursively, connecting ancient patterns to modern algebra.

So here is the big question this lesson addresses: if a sequence is just a list of numbers, why do mathematicians call it a function? And what does it mean when a sequence is defined recursively — that is, when each term depends on the terms that came before it? Let's find out.

SECTION 2

Core Principles & Definitions

Before we dive into examples, let's nail down the key ideas you need to know. A sequence is an ordered list of numbers, and each number in the list is called a term. When we treat a sequence as a function, we use the position number (1st, 2nd, 3rd…) as the input and the term value as the output.

Sequence = Function

A sequence is a function whose domain (set of inputs) is a subset of the integers — typically {0, 1, 2, 3, …} or {1, 2, 3, …}. Each input gives exactly one output.

Explicit Formula

An explicit formula lets you calculate any term directly from its position number n. For example, f(n) = 2n gives the even numbers: 2, 4, 6, 8, …

Recursive Formula

A recursive formula defines each term using one or more previous terms, plus one or more starting values called initial conditions.

Initial Conditions

Recursive formulas need a starting point. The initial condition(s) tell you the value of the first term (or first few terms) so the recursion can get going.

Domain is Integers

Unlike most functions you graph on a coordinate plane, a sequence's domain is only whole numbers. You can't find the "2.5th" term of a sequence — only the 1st, 2nd, 3rd, and so on.

✦ KEY TAKEAWAY

Think of a sequence like a playlist on your phone. Each song has a position number (track 1, track 2, track 3…), and each position maps to exactly one song — that's a function! A recursive playlist would be one where each song is chosen based on the song before it, like a DJ picking the next track to match the mood of the current one.

SECTION 3

Visualizing Sequences as Functions

When we plot a sequence on the coordinate plane, we place the position number n on the horizontal axis and the term value f(n) on the vertical axis. Because the domain is only whole numbers, the graph is a set of discrete dots rather than a smooth, continuous curve. The diagram below shows two sequences side by side: an arithmetic sequence where each term increases by a constant amount, and the Fibonacci sequence where each term is the sum of the two before it.

Left: the arithmetic sequence f(n) = 3n forms a straight-line pattern of dots. Right: the Fibonacci sequence curves upward rapidly because each term adds the two before it. Notice both graphs are discrete dots, not continuous lines, since the domain is only whole numbers.

Notice a key difference between the two graphs. On the left, the arithmetic sequence's dots line up perfectly, which makes sense because each term grows by the same amount (3). On the right, the Fibonacci sequence starts slowly but then shoots upward, because each new term is the sum of two growing previous terms. Both graphs confirm the function idea: for every position n, there is exactly one dot — exactly one output.

SECTION 4

Mathematical Framework

Let's now look at the formal notation used to describe sequences as functions. There are two main ways to write the rule for a sequence: an explicit (closed-form) formula and a recursive formula. Both define the same function, but they give you the output in different ways.

EXPLICIT FORMULA (ARITHMETIC SEQUENCE)

f(n) = a₁ + d × (n − 1)

a₁ = first term, d = common difference, n = position number. This lets you jump directly to any term.

RECURSIVE FORMULA (ARITHMETIC SEQUENCE)

f(1) = a₁, f(n) = f(n − 1) + d for n ≥ 2

Start with the first term f(1) = a₁. To find any later term, add d to the previous term. You must work forward one step at a time.

RECURSIVE FORMULA (FIBONACCI SEQUENCE)

f(0) = 1, f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1

The Fibonacci sequence needs two initial conditions because each term depends on the two terms before it. There is no simple explicit formula.

EXPLICIT FORMULA (GEOMETRIC SEQUENCE)

f(n) = a₁ × r⁽ⁿ⁻¹⁾

a₁ = first term, r = common ratio, n = position number. Geometric sequences multiply by the same factor each time.

The choice between explicit and recursive depends on the situation. If you need the 100th term right away, an explicit formula is much faster because you plug in n = 100 and compute. But if a sequence is naturally defined by how each term relates to the ones before it — like the Fibonacci sequence — then a recursive definition is the natural and sometimes only option.

SECTION 5

Types of Sequences & Their Formulas

Now that you know the two ways to write a sequence's rule, let's organize the most important sequence types you'll encounter in Algebra 1. The diagram below shows how each type connects to its explicit and recursive forms.

The Sequence Family Tree. Arithmetic and geometric sequences each have both explicit and recursive forms. The Fibonacci sequence is naturally recursive — there is no simple closed-form formula at this level.

Common Sequence Types in Algebra 1

Sequence Type	Example	Pattern Rule	Explicit or Recursive?
Arithmetic	2, 5, 8, 11, 14, …	Add 3 each time (d = 3)	Both available
Geometric	3, 6, 12, 24, 48, …	Multiply by 2 each time (r = 2)	Both available
Fibonacci	1, 1, 2, 3, 5, 8, 13, …	Add the two previous terms	Recursive (naturally)
Squares	1, 4, 9, 16, 25, …	f(n) = n²	Explicit is simpler

SECTION 6

Worked Example: Building the Fibonacci Sequence

Let's work through a complete example using the Fibonacci sequence's recursive definition to find the first eight terms. Then we'll show why the sequence truly behaves like a function.

Find the first 8 terms of the Fibonacci sequence

Step 1 — Write the recursive rule and initial conditionsThe Fibonacci sequence is defined by: f(0) = 1, f(1) = 1, and f(n + 1) = f(n) + f(n − 1) for n ≥ 1. We know the first two terms: f(0) = 1 and f(1) = 1.

f(0) = 1, f(1) = 1

Step 2 — Calculate f(2)Using the formula with n = 1: f(2) = f(1) + f(0) = 1 + 1.

f(2) = 2

Step 3 — Calculate f(3)Now use n = 2: f(3) = f(2) + f(1) = 2 + 1.

f(3) = 3

Step 4 — Calculate f(4) through f(7)Continue the same process: f(4) = f(3) + f(2) = 3 + 2 = 5. Then f(5) = f(4) + f(3) = 5 + 3 = 8. Then f(6) = f(5) + f(4) = 8 + 5 = 13. Finally, f(7) = f(6) + f(5) = 13 + 8 = 21.

f(4) = 5, f(5) = 8, f(6) = 13, f(7) = 21

Step 5 — Verify it's a functionCheck: does each input n give exactly one output? Yes! For n = 0, the output is always 1. For n = 5, the output is always 8. No input produces two different values. The domain is {0, 1, 2, 3, 4, 5, 6, 7}, a subset of the integers.

Complete sequence: 1, 1, 2, 3, 5, 8, 13, 21

💡 Why does recursion need initial conditions?

Without the starting values f(0) = 1 and f(1) = 1, the rule f(n + 1) = f(n) + f(n − 1) has nothing to work with. It's like saying "the next step is two steps back, added together" — if you haven't taken any steps yet, you're stuck! The initial conditions are the launch pad.

SECTION 7

Explicit vs. Recursive: Strengths & Limitations

You might wonder: if explicit formulas let you jump to any term instantly, why bother with recursion at all? The answer is that some sequences can only be described naturally through recursion. Let's compare the two approaches side by side.

Comparing Explicit and Recursive Formulas

Feature	Explicit Formula	Recursive Formula
Speed to find the 100th term	Fast — plug in n = 100 directly	Slow — must compute all 99 terms before it
Flexibility	Works well for arithmetic and geometric sequences	Can describe any sequence, even complex ones like Fibonacci
Ease of understanding	Sometimes harder to derive	Often mirrors how the pattern naturally works
Needs initial conditions?	No — the formula is self-contained	Yes — must state starting term(s)
Best for	Finding specific terms far into the sequence	Modeling processes that build on previous results

✦ KEY TAKEAWAY

Think of it like giving directions to your house. An explicit formula is like giving someone a GPS address — they can go straight there. A recursive formula is like turn-by-turn directions — "from where you are now, turn left, go 2 blocks, then turn right." Both get you there, but each is better in different situations.

SECTION 8

Connection to Advanced Topics

Understanding sequences as functions and mastering recursion opens doors to many advanced topics in mathematics and computer science. The table below previews how the ideas in this lesson connect to what you'll study later.

From Algebra 1 to Advanced Mathematics

This Lesson (Algebra 1)	Where It Leads
Sequences have integer domains	In Precalculus, you study series — the sum of all terms in a sequence — and summation notation (Σ)
Recursive formulas build term by term	In computer science, recursion is a fundamental programming technique used in algorithms and data structures
Arithmetic sequences grow by a constant	In Algebra 2, you connect arithmetic sequences to linear functions and use them to model real-world growth
Geometric sequences grow by a constant ratio	In Precalculus and Calculus, geometric sequences connect to exponential functions, compound interest, and infinite series
The Fibonacci sequence grows rapidly	In advanced math, the Fibonacci sequence connects to the golden ratio (φ ≈ 1.618), fractal geometry, and nature's patterns

The core insight from this lesson — that a sequence is a function with an integer domain — is a bridge between the patterns you see in everyday life and the formal mathematical tools you'll use for years to come. Whether you go into science, engineering, finance, or computer programming, recursive thinking will be one of your most powerful problem-solving skills.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain in your own words why a sequence is a function. What is the domain? What is the range? Why can't you find the "3.5th" term of a sequence?

PROBLEM 2 — BASIC CALCULATION

A sequence is defined recursively by f(1) = 4 and f(n) = f(n − 1) + 7 for n ≥ 2. Find the first five terms of the sequence.

PROBLEM 3 — INTERMEDIATE

Consider the sequence 5, 10, 20, 40, 80, … (a) Write a recursive formula for this sequence. (b) Write an explicit formula. (c) Use your explicit formula to find the 10th term.

PROBLEM 4 — APPLIED

A biologist observes a bacteria colony. On day 0 there are 100 bacteria. Each day, the population increases by 15% of the previous day's population. (a) Write a recursive formula for the population P(n) on day n. (b) Find the population on day 3 (round to the nearest whole number). (c) Is this sequence arithmetic, geometric, or neither?

PROBLEM 5 — CRITICAL THINKING

A new sequence is defined recursively by g(1) = 2, g(2) = 5, and g(n) = g(n − 1) − g(n − 2) for n ≥ 3. (a) Find the first eight terms. (b) What pattern do you notice? (c) Explain why this sequence eventually repeats and state the length of the repeating cycle.

SUMMARY

Lesson Summary

In this lesson, you learned that a sequence is a function whose domain is a subset of the integers — each whole-number position maps to exactly one term value. You explored two ways to define a sequence: an explicit formula that lets you jump directly to any term, and a recursive formula that builds each term from the ones before it. Both arithmetic and geometric sequences can be written either way.

The Fibonacci sequence — defined by f(0) = 1, f(1) = 1, and f(n + 1) = f(n) + f(n − 1) — is the classic example of a sequence that is naturally recursive. Every recursive formula requires initial conditions — the starting values that get the pattern going. When you graph a sequence, the result is a set of discrete dots (not a continuous curve), reflecting its integer domain. Mastering these ideas prepares you for series, advanced algebra, and recursive thinking in computer science.