Modeling with Sequences — I can use sequences to model...

SECTION 1

Historical Context & Motivation

Long before algebra textbooks existed, people noticed that many real-world situations follow predictable, repeating patterns. Ancient farmers tracked how their harvests changed from season to season, merchants calculated interest on loans, and architects designed staircases with equal step heights. All of these situations share a common thread: repeated change — the same operation happening over and over again. The mathematical tool that captures this idea is called a sequence, an ordered list of numbers generated by a consistent rule.

~300 BCE

Euclid's Elements

Euclid studied geometric progressions and proved early results about sums of sequences, laying the groundwork for the algebra of repeated multiplication.

~500 CE

Aryabhata & Indian Mathematics

Indian mathematician Aryabhata developed formulas for the sum of arithmetic sequences, enabling faster calculations for trade and astronomy.

1202

Fibonacci's Liber Abaci

Leonardo of Pisa introduced Europe to the Hindu-Arabic numeral system and explored the famous Fibonacci sequence, showing how sequences model growth in nature.

1600s

Compound Interest & Finance

European bankers formalized geometric sequences to compute compound interest, making sequences a cornerstone of modern finance.

Today

Sequences Everywhere

From monthly car payments to computer algorithms that process data in steps, sequences remain one of the most practical tools in mathematics.

The central question this lesson addresses is straightforward: when you see a situation where the same change happens again and again — adding $50 to your savings each week, climbing stairs of equal height, or a population doubling every year — how do you write a formula that predicts any value in the pattern without listing every single term? That is the power of modeling with sequences.

SECTION 2

Core Principles & Definitions

Before diving into formulas, you need a solid foundation in the vocabulary and ideas that make sequences work. A sequence is simply an ordered list of numbers, called terms, that follow a definite pattern or rule. The position of each term matters — the first term, second term, third term, and so on. Two types of sequences dominate real-world modeling: arithmetic and geometric.

Arithmetic Sequence

Each term is found by adding a constant value, called the common difference (d), to the previous term. Example: 3, 7, 11, 15, … where d = 4.

Geometric Sequence

Each term is found by multiplying the previous term by a constant value, called the common ratio (r). Example: 2, 6, 18, 54, … where r = 3.

First Term (a₁)

The starting value of the sequence. Every formula needs this anchor point to generate the rest of the list.

Recursive vs. Explicit Rules

A recursive rule tells you the next term from the current one. An explicit rule lets you jump directly to any term using its position number n.

✦ KEY TAKEAWAY

Think of an arithmetic sequence like walking up a staircase where every step is the same height — you always add the same amount. A geometric sequence is more like a chain of photocopies set to enlarge — each copy is multiplied by the same factor. Knowing which operation repeats (addition or multiplication) is the key to choosing the right model.

SECTION 3

Visual Explanation — Step Patterns

One of the best ways to understand arithmetic sequences is to see them as growing step patterns. Imagine building figures out of tiles where each new figure adds a consistent number of tiles. The diagram below shows three figures in a tile pattern, with each step adding 3 tiles to the previous figure.

Each figure adds 3 new tiles (shown in a different color). The cyan tiles represent the original 2, the violet tiles are the first addition of 3, and the pink tiles are the second addition of 3. This visual makes the common difference d = 3 easy to spot.

Notice how each color group in the diagram corresponds to a specific addition of d = 3 tiles. The original cyan tiles never disappear — they remain as the first term a₁. Every new figure tacks on one more group, so by Figure n you have added d exactly (n − 1) times. This visual reasoning leads directly to the explicit formula for arithmetic sequences.

SECTION 4

Mathematical Framework

Now that you can see sequences visually, let's formalize the math. Each type of sequence — arithmetic and geometric — has both a recursive rule (how to get the next term from the current one) and an explicit rule (how to jump to any term by its position number n).

Arithmetic Sequences

RECURSIVE RULE — ARITHMETIC

aₙ = aₙ₋₁ + d

aₙ = current term; aₙ₋₁ = previous term; d = common difference (the constant amount added each step).

EXPLICIT RULE — ARITHMETIC

aₙ = a₁ + (n − 1) × d

a₁ = first term; n = term number (position). This formula lets you find any term without computing all the ones before it.

Geometric Sequences

RECURSIVE RULE — GEOMETRIC

aₙ = aₙ₋₁ × r

aₙ = current term; aₙ₋₁ = previous term; r = common ratio (the constant factor you multiply by).

EXPLICIT RULE — GEOMETRIC

aₙ = a₁ × r⁽ⁿ⁻¹⁾

a₁ = first term; n = term number; r is raised to the power (n − 1) because the first term is not multiplied at all.

💡 How to Tell Them Apart

Subtract consecutive terms: if the differences are constant, the sequence is arithmetic. Divide consecutive terms: if the ratios are constant, the sequence is geometric. If neither is constant, it's another type of sequence entirely.

SECTION 5

Arithmetic vs. Geometric — A Side-by-Side View

The difference between arithmetic and geometric sequences becomes crystal clear when you graph them. An arithmetic sequence produces points that line up in a straight line, while a geometric sequence produces a curve that rises (or falls) faster and faster. The diagram below plots both types on the same coordinate plane so you can compare their growth.

The cyan dots form a straight line (arithmetic growth with d = 5), while the pink dots form a curve that accelerates upward (geometric growth with r ≈ 2). For small n the sequences are close, but geometric growth eventually dominates.

Key differences between arithmetic and geometric sequences

Feature	Arithmetic Sequence	Geometric Sequence
Operation	Repeated addition	Repeated multiplication
Constant	Common difference d	Common ratio r
Graph shape	Straight line (linear)	Curve (exponential)
Real-world example	Saving $50 per week	Bacteria doubling every hour

SECTION 6

Worked Example — Monthly Savings Plan

Suppose you start a summer savings plan. You deposit $120 in the first month and add $45 every month after that. How much will your monthly deposit be in month 10? Let's model this with an arithmetic sequence.

Monthly Savings Deposit — Finding the 10th Term

Step 1 — Identify the type of sequenceYou are adding a fixed $45 each month, which means the deposits follow an arithmetic sequence. The repeating operation is addition, not multiplication.

Step 2 — Extract the key valuesFirst term: a₁ = 120 (the deposit in month 1). Common difference: d = 45 (the amount added each month). Term number: n = 10.

a₁ = 120, d = 45, n = 10

Step 3 — Write the explicit formulaThe explicit formula for an arithmetic sequence is aₙ = a₁ + (n − 1) × d. We substitute our values.

a₁₀ = 120 + (10 − 1) × 45

Step 4 — Simplify inside the parenthesesCalculate (10 − 1) first, which equals 9. Then multiply: 9 × 45 = 405.

a₁₀ = 120 + 405

Step 5 — Compute the final answerAdd 120 + 405 = 525. So in month 10, your deposit will be $525. This makes sense — you started at $120 and added $45 nine more times.

a₁₀ = $525

✔ Check Your Reasoning

A quick sanity check: the first few terms are 120, 165, 210, 255, … Each jump is exactly 45 — that confirms d is correct. And nine jumps of 45 equal 405, which added to 120 gives 525. Always verify a few terms when building a sequence model.

SECTION 7

Strengths & Limitations of Sequence Models

Sequences are powerful tools, but no model is perfect. Understanding when a sequence model works well — and when it breaks down — will help you choose the right approach for a given problem.

When sequence models shine and when they fall short

Strengths	Limitations
Predict any term without listing all earlier terms (using the explicit formula).	Only works when the change truly is constant — many real situations involve varying rates of change.
Easy to identify: just check if differences or ratios are constant.	Geometric sequences can grow unrealistically fast (e.g., infinite bacteria in reality hit a resource limit).
Applicable to finance (payments), construction (step patterns), biology (population growth), and more.	Sequences model discrete (step-by-step) situations; continuous change often needs functions or calculus.
Formulas are algebraically simple and accessible at the Algebra 1 level.	Mixed patterns (e.g., add 5, then multiply by 2, then add 5…) do not fit a simple arithmetic or geometric model.

✦ KEY TAKEAWAY

A sequence model is like a map of a straight highway — it tells you exactly where you'll be at every mile marker as long as the road stays straight. But if the road curves, the map becomes inaccurate. Similarly, sequence formulas work perfectly when the rate of change is constant, but break down when the rate itself starts changing.

SECTION 8

Connection to Functions and Advanced Topics

Sequences are actually a stepping stone to one of the biggest ideas in algebra: functions. Every explicit sequence formula can be thought of as a function where the input is the term number n. Arithmetic sequences connect to linear functions, and geometric sequences connect to exponential functions. Recognizing this link will make your transition to these topics smoother.

How sequence concepts map to future algebra and beyond

Concept Now	Where It Leads
Arithmetic sequence: aₙ = a₁ + (n − 1)d	Linear function: f(x) = mx + b, where m = d and b = a₁ − d
Geometric sequence: aₙ = a₁ × r⁽ⁿ⁻¹⁾	Exponential function: f(x) = a × bˣ, used for compound interest and population models
Common difference d	Slope of a line (rate of change)
Common ratio r	Growth factor in exponential models
Recursive formula (next from previous)	Recursive functions in computer science and advanced math

Later in your math journey, you'll encounter series (summing the terms of a sequence), sigma notation, and eventually infinite sequences in calculus. The foundation you build right now — identifying the type of change, writing a formula, and using it to predict — carries forward into all of those advanced topics.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A staircase has steps that are each 7 inches tall. Is the total height after n steps modeled by an arithmetic or geometric sequence? Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

An arithmetic sequence has a₁ = 8 and d = 6. Find the 15th term using the explicit formula.

PROBLEM 3 — INTERMEDIATE

A geometric sequence starts at 4 and has a common ratio of 3. Write the explicit formula and find the 6th term.

PROBLEM 4 — APPLIED

You sign up for a streaming service that costs $12 the first month but increases by $1.50 each month due to premium add-ons. How much will you pay in month 12? What is the total amount paid over all 12 months?

PROBLEM 5 — CRITICAL THINKING

A sequence begins: 5, 15, 45, 135, … Another sequence begins: 5, 20, 35, 50, … At what term number does the geometric sequence first exceed 1,000? At what term number does the arithmetic sequence first exceed 1,000? What does this comparison reveal about arithmetic vs. geometric growth?

SUMMARY

Lesson Summary

A sequence is an ordered list of numbers following a rule. When the same change repeats — adding a constant (arithmetic sequence, common difference d) or multiplying by a constant (geometric sequence, common ratio r) — you can write an explicit formula to predict any term directly. The arithmetic explicit formula is aₙ = a₁ + (n − 1) × d and the geometric explicit formula is aₙ = a₁ × r⁽ⁿ⁻¹⁾.

Use sequences to model real-world situations like step patterns (adding tiles), monthly payments (adding fixed amounts), and population growth (multiplying by a factor). Arithmetic sequences graph as straight lines and connect to linear functions, while geometric sequences graph as exponential curves and connect to exponential functions. Identifying the repeating operation — addition or multiplication — is the first and most important step in building your sequence model.