Write Explicit or Recursive Functions - Algebra 1

SECTION 1

Historical Context & Motivation

Humans have been fascinated by patterns in numbers for thousands of years. Long before modern algebra existed, mathematicians noticed that certain sequences—like the counting numbers 1, 2, 3, 4, … or the squares 1, 4, 9, 16, …—follow predictable rules. The challenge was finding a way to describe those rules precisely so that anyone could use them to predict future values.

Over time, two main strategies emerged. One approach says, "Give me the position number, and I'll tell you the value directly." The other says, "Give me the previous value, and I'll tell you how to get the next one." These two approaches are what we now call explicit functions and recursive functions. Understanding both gives you a complete toolkit for describing sequences and patterns.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid described number patterns using step-by-step processes. His method for finding the greatest common divisor is one of the earliest recorded recursive algorithms in history.

1202 • Fibonacci's Liber Abaci

Leonardo of Pisa (Fibonacci) introduced a famous sequence—1, 1, 2, 3, 5, 8, …—where each term is the sum of the two before it. This is a classic example of a recursive definition.

1600s • Rise of Algebraic Notation

Mathematicians like Descartes and Fermat began using variables and formulas to express patterns directly. This made explicit formulas a standard tool for describing sequences.

1900s • Computer Science & Recursion

With the birth of computers, recursion became a cornerstone of programming. Functions that call themselves to solve smaller sub-problems power everything from search engines to video game graphics.

Today, CCSS standard F-BF.1.a asks you to do exactly what these mathematicians did: look at a real-world situation or a number pattern and determine whether an explicit expression, a recursive process, or a set of calculation steps best captures the pattern. Let's learn how.

SECTION 2

Core Principles & Definitions

Before we dive into examples, let's nail down the key vocabulary. A sequence is an ordered list of numbers that follows a pattern. Each number in the list is called a term. We use the notation aₙ to mean "the term at position n." So a₁ is the first term, a₂ is the second term, and so on.

Explicit Function

A formula that lets you find any term directly from its position number n. Example: aₙ = 3n + 1. Plug in n = 50, and you instantly get a₅₀ = 151.

Recursive Function

A rule that tells you how to get the next term from the previous one. You always need a starting value (the first term). Example: a₁ = 4, aₙ = aₙ₋₁ + 3.

Arithmetic Sequence

A sequence where you add the same number (the common difference, d) each time. Examples: 2, 5, 8, 11, … (d = 3) or 10, 7, 4, 1, … (d = −3).

Geometric Sequence

A sequence where you multiply by the same number (the common ratio, r) each time. Examples: 3, 6, 12, 24, … (r = 2) or 100, 50, 25, … (r = 0.5).

Initial Condition

The starting value required by every recursive function. Without it, the recursion has nothing to build from. Think of it as the first domino in a chain.

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KEY TAKEAWAY

Think of an explicit function like a GPS—it takes you directly to any destination (term) without stopping along the way. A recursive function is like driving turn by turn—each instruction depends on where you currently are, and you have to start from home (the initial condition). Both get you there, but one is faster for distant terms.

SECTION 3

Visual Explanation: Explicit vs. Recursive

The diagram below shows the arithmetic sequence 4, 7, 10, 13, 16, … in two ways. On the left, the explicit approach draws a direct arrow from each position number n to its term value. On the right, the recursive approach shows how each term is built by adding 3 to the term before it. Notice how the explicit path is a single hop, while the recursive path chains through every preceding term.

Left: The explicit function aₙ = 3n + 1 maps each position n directly to its value (single arrow). Right: The recursive function a₁ = 4, aₙ = aₙ₋₁ + 3 builds each term from the previous one (chain of arrows).

Notice the key difference: with the explicit formula, if someone asks "What is the 100th term?" you just plug in n = 100 and compute 3(100) + 1 = 301. With the recursive formula, you would theoretically need to calculate all 99 terms before it to reach the 100th. Both formulas describe the same sequence, but they reveal different aspects of its structure.

SECTION 4

Mathematical Framework

Let's formalize the two types of formulas for the two most common sequence families: arithmetic and geometric.

Arithmetic Sequences

EXPLICIT — ARITHMETIC

aₙ = a₁ + (n − 1) × d

where a₁ = first term, n = position number, and d = common difference (the amount added each step).

RECURSIVE — ARITHMETIC

a₁ = (first term), aₙ = aₙ₋₁ + d

You must state the initial condition (the value of a₁) plus the rule that each new term equals the previous term plus d.

Geometric Sequences

EXPLICIT — GEOMETRIC

aₙ = a₁ × r⁽ⁿ⁻¹⁾

where a₁ = first term, r = common ratio (the number you multiply by each step), and n = position number.

RECURSIVE — GEOMETRIC

a₁ = (first term), aₙ = aₙ₋₁ × r

Again, you need the initial condition plus the rule that each new term is the previous term multiplied by r.

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Watch Out!

A recursive function without an initial condition is incomplete. Writing "aₙ = aₙ₋₁ + 5" alone doesn't tell you where the sequence starts. The sequence could be 2, 7, 12, 17, … or 100, 105, 110, 115, … — both follow the same recursive rule! Always include a₁ = ___.

SECTION 5

Choosing Between Explicit and Recursive

How do you decide which type of function to write? It depends on the context. Sometimes a problem naturally suggests one form over the other. The diagram below walks you through a decision-making process.

Follow this flowchart when a problem asks you to write a function from a real-world context. Start by asking whether the situation describes how each new value relates to the previous one (recursive) or whether you can jump directly to any position (explicit).

Clue in the Problem	Form to Use	Example Wording
"Each week, the amount increases by $50"	Recursive (arithmetic)	a₁ = 200, aₙ = aₙ₋₁ + 50
"The population doubles every year"	Recursive (geometric)	a₁ = 500, aₙ = 2 × aₙ₋₁
"What is the value at week 52?"	Explicit (more efficient)	aₙ = 200 + (n − 1) × 50
"Write a rule for the nth term"	Explicit	aₙ = a₁ + (n − 1)d or aₙ = a₁ × r⁽ⁿ⁻¹⁾

Common problem clues and which function form they suggest

SECTION 6

Worked Example

Let's work through a context-based problem from start to finish using both approaches.

ℹ

Problem

A movie streaming service has 2,000 subscribers in January. Each month, 350 new subscribers join. Write (a) a recursive function and (b) an explicit function to model the number of subscribers after n months. Then (c) find the number of subscribers after 12 months.

Solution

1
Step 1 — Identify the type of sequence

The number of subscribers increases by the same amount (350) every month. A constant addition means this is an arithmetic sequence with first term a₁ = 2,000 and common difference d = 350.

a₁ = 2,000, d = 350
2
Step 2 — Write the recursive function

For the recursive form, state the initial condition and the rule: each month's subscriber count equals last month's count plus 350.

a₁ = 2,000, aₙ = aₙ₋₁ + 350
3
Step 3 — Write the explicit function

Use the arithmetic explicit formula aₙ = a₁ + (n − 1) × d. Substitute a₁ = 2,000 and d = 350: aₙ = 2,000 + (n − 1) × 350 Simplify: aₙ = 2,000 + 350n − 350 = 350n + 1,650.

aₙ = 350n + 1,650
4
Step 4 — Find the number of subscribers at month 12

Plug n = 12 into the explicit formula: a₁₂ = 350(12) + 1,650 = 4,200 + 1,650 = 5,850.

a₁₂ = 5,850 subscribers
5
Step 5 — Verify with the recursive formula

Quick check: a₁ = 2,000, a₂ = 2,350, a₃ = 2,700 … the pattern adds 350 each time. After 11 additions from a₁ we get 2,000 + 11 × 350 = 2,000 + 3,850 = 5,850. ✓ Both methods agree.

✓ Confirmed: 5,850

SECTION 7

Explicit vs. Recursive — Strengths & Limitations

Neither form is "better" in all situations. Each has strengths that make it the natural choice depending on what you need to do.

Feature	Explicit Function	Recursive Function
Finding a distant term (e.g., a₁₀₀)	Fast—plug n = 100 directly into the formula.	Slow—you must calculate all 99 previous terms.
Ease of writing from context	May require algebra to simplify.	Often easier—the context literally tells you "add 5 each time."
Shows the overall pattern	Yes—you can see how aₙ depends on n globally.	No—focuses on local step-to-step changes.
Requires an initial condition	No—it's built into the formula.	Yes—always needs a₁ (or a₀) stated.
Best for computer programs / spreadsheets	Good for random access (jump to any row).	Great for looping (fill cells one at a time).

Comparison of explicit and recursive function forms

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KEY TAKEAWAY

Think of it like a recipe versus an ingredient list. The recursive function is the step-by-step recipe: "Take what you have and add 3 cups of flour." The explicit function is the total ingredient list: "For n batches, you need 3n cups of flour plus 1 cup of sugar." A skilled baker (mathematician!) can use either one depending on the situation.

SECTION 8

Connection to Advanced Topics

The ideas you've learned here form the foundation for more advanced work in Algebra 2, Pre-Calculus, and beyond. Understanding explicit and recursive definitions prepares you for topics like series and summation notation, exponential growth models, and even mathematical induction (a proof technique that relies on recursive thinking).

What You Learn Now	Where It Leads
Arithmetic explicit formula: aₙ = a₁ + (n − 1)d	Linear functions y = mx + b (d becomes the slope, a₁ − d becomes the y-intercept)
Geometric explicit formula: aₙ = a₁ × r⁽ⁿ⁻¹⁾	Exponential functions y = a × bˣ (used in finance, biology, and physics)
Recursive definitions with initial conditions	Recursion in computer science, difference equations, mathematical induction
Summing terms of a sequence	Series, sigma notation (Σ), and eventually integral calculus

How today's concepts connect to future coursework

In particular, the Fibonacci sequence (1, 1, 2, 3, 5, 8, …) is easy to define recursively—aₙ = aₙ₋₁ + aₙ₋₂ with a₁ = 1 and a₂ = 1—but its explicit formula involves irrational numbers and the golden ratio. This shows that not every recursive pattern translates neatly into a simple explicit formula, which is part of why mathematicians value both representations.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In your own words, explain the difference between an explicit function and a recursive function. Why does a recursive function always need an initial condition?

Show answer

An explicit function gives you a formula that directly calculates any term from its position number n—you don't need to know any other terms. A recursive function tells you how to get each term from the previous term, so it builds the sequence one step at a time. The initial condition (the value of a₁) is required because, without a starting point, the recursive rule has nothing to build from. For example, "add 5 each time" could describe infinitely many different sequences depending on where you start.

Question 2: BASIC CALCULATION

Consider the arithmetic sequence 6, 11, 16, 21, 26, … Write (a) a recursive function and (b) an explicit function for this sequence. Then find the 20th term.

Show answer

First, identify a₁ = 6 and the common difference d = 11 − 6 = 5. (a) Recursive: a₁ = 6, aₙ = aₙ₋₁ + 5. (b) Explicit: aₙ = 6 + (n − 1) × 5 = 5n + 1. For the 20th term, substitute n = 20 into the explicit formula: a₂₀ = 5(20) + 1 = 101. You can verify: starting at 6 and adding 5 a total of 19 times gives 6 + 95 = 101. ✓

Question 3: INTERMEDIATE

A geometric sequence begins 3, 12, 48, 192, … (a) Write a recursive function. (b) Write an explicit function. (c) What is the 8th term?

Show answer

The common ratio is r = 12 ÷ 3 = 4, and a₁ = 3. (a) Recursive: a₁ = 3, aₙ = 4 × aₙ₋₁. (b) Explicit: aₙ = 3 × 4⁽ⁿ⁻¹⁾. (c) For n = 8: a₈ = 3 × 4⁷ = 3 × 16,384 = 49,152. Notice how the explicit form is essential here—computing all 8 terms recursively would require seven multiplications, whereas the explicit formula gives the answer in one step (plus evaluating 4⁷).

Question 4: APPLIED

A concert venue sells tickets and tracks total revenue. On day 1, they earn $1,200. Each subsequent day, they earn $75 less than the day before (as demand drops). (a) Write a recursive function for daily revenue. (b) Write an explicit function. (c) On which day does daily revenue first drop to $0 or below?

Show answer

Here a₁ = 1,200 and d = −75 (the revenue decreases). (a) Recursive: a₁ = 1,200, aₙ = aₙ₋₁ − 75. (b) Explicit: aₙ = 1,200 + (n − 1)(−75) = 1,200 − 75n + 75 = 1,275 − 75n. (c) Set aₙ ≤ 0: 1,275 − 75n ≤ 0, so 75n ≥ 1,275, meaning n ≥ 17. On day 17, revenue = 1,275 − 75(17) = 1,275 − 1,275 = 0. The revenue first hits $0 on day 17. This kind of model helps event planners decide how many days to sell tickets.

Question 5: CRITICAL THINKING

A sequence is defined recursively as: a₁ = 2, aₙ = 3aₙ₋₁ − 4. (a) List the first five terms. (b) Is this sequence arithmetic, geometric, or neither? Justify your answer. (c) Can you write an explicit formula for this sequence? (Hint: try to spot a pattern in the terms or rewrite the recursive rule.)

Show answer

a₁ = 2, a₂ = 3(2) − 4 = 2, a₃ = 3(2) − 4 = 2, a₄ = 2, a₅ = 2. Every term equals 2! This is technically a constant sequence. It is arithmetic (d = 0) and could also be viewed as geometric (r = 1). (b) It's arithmetic with d = 0. (c) The explicit formula is simply aₙ = 2 for all n. This problem illustrates an important lesson: always compute a few terms before assuming a recursive rule produces a "complicated" sequence. The fixed point occurs because 2 satisfies the equation x = 3x − 4, giving x = 2. Any starting value other than 2 would produce a very different, non-constant sequence.

SUMMARY

Lesson Summary

Summary

In this lesson, you learned two fundamental ways to describe number patterns. An explicit function uses a formula like aₙ = a₁ + (n − 1)d (arithmetic) or aₙ = a₁ × r⁽ⁿ⁻¹⁾ (geometric) to jump directly to any term. A recursive function defines each term in relation to the previous one and always requires an initial condition (the value of a₁).

When a problem describes step-by-step changes ("each week, add 50"), a recursive form is the natural choice. When you need to find a distant term quickly or see the overall pattern, an explicit form is more efficient. The key to CCSS F-BF.1.a is reading the context carefully, identifying whether the sequence is arithmetic or geometric, extracting the common difference (d) or common ratio (r), and writing the appropriate function. With practice, you'll be able to move fluidly between both representations.