Arithmetic & Geometric Sequences: Recursive & Explicit...

Section 1

Where Sequences Came From

Long before anyone called them "sequences," people noticed patterns in numbers. A farmer who added the same number of sheep to his flock each year, or a merchant who doubled the weight of rice on each square of a chessboard — both were working with ideas that would eventually become the arithmetic and geometric sequences you'll study today. Understanding where these patterns first appeared helps you see why mathematicians created special formulas for them.

≈ 1700 BCE

Ancient Babylonian clay tablets include tables of numbers that grow by a constant amount — the earliest known examples of arithmetic sequences. These tables helped scribes calculate areas of fields and volumes of grain stores.

≈ 300 BCE

Euclid, the Greek mathematician, studied geometric proportions in his masterwork Elements. He showed how to sum terms in a geometric progression, which became the basis for interest calculations and population models centuries later.

≈ 800 CE

The legend of Sissa ibn Dahir and the chessboard: place 1 grain of wheat on the first square, 2 on the second, 4 on the third, doubling each time. The total — a geometric series — exceeds all the wheat in the world, illustrating exponential growth.

1800s

Carl Friedrich Gauss, as a schoolboy, famously summed the integers from 1 to 100 in seconds by recognizing it as an arithmetic sequence. His shortcut became the standard formula still taught today: S = n(n + 1) / 2.

Today

Arithmetic and geometric sequences appear everywhere — from calculating monthly loan payments and predicting population growth to programming loops in computer science and modeling radioactive decay in physics.

The central question these historical discoveries point to is: if you know a pattern's rule, can you predict any term in the sequence — even the 100th or the 1,000th — without listing them all? The answer is yes, and the tools that make it possible are recursive formulas and explicit formulas.

Section 2

Core Definitions & Principles

Before we build formulas, let's nail down the vocabulary. A sequence is simply an ordered list of numbers, and each number in the list is called a term. We label terms using a subscript: a₁ is the first term, a₂ is the second, and aₙ is the nth term (the general term). The two families of sequences you need to know are defined by their specific growth rules.

Arithmetic Sequence

Each term is found by adding the same fixed number to the previous term. That fixed number is called the common difference, written d. Example: 3, 7, 11, 15, … (d = 4).

Geometric Sequence

Each term is found by multiplying the previous term by the same fixed number. That number is called the common ratio, written r. Example: 2, 6, 18, 54, … (r = 3).

Recursive Formula

Tells you how to get the next term from the previous one. You always need a starting value (a₁) and a rule like "add d" or "multiply by r." Think of it as a set of step-by-step directions.

Explicit Formula

Tells you the value of any term directly from its position number n. You don't need any earlier terms — just plug in n. Think of it as a shortcut that teleports you straight to the answer.

✦ Key Takeaway

Imagine you're climbing stairs. A recursive formula is like knowing "take one more step up from wherever you are" — you always need your current position. An explicit formula is like having an elevator that takes you directly to any floor — just press the button for the floor number. Both get you there, but the elevator is faster when you want the 200th floor.

Section 3

Visualizing Both Sequence Types

A picture is worth a hundred terms. The diagram below plots the first six terms of an arithmetic sequence (cyan dots) and a geometric sequence (violet dots) on the same coordinate plane. Notice how the arithmetic sequence grows in a straight line while the geometric sequence curves upward.

Comparison of arithmetic (linear) and geometric (exponential) sequences starting at a₁ = 2.

Both sequences start at the same value (a₁ = 2), but their growth rules produce very different shapes. The arithmetic sequence adds d = 3 each time, which creates a straight line — constant growth, just like a linear function. The geometric sequence multiplies by r = 2 each time, which creates a curve that bends upward — exponential growth. By term 6, the geometric sequence has already reached 64 while the arithmetic one is only at 17. This visual difference is key: if you see data points forming a line, think arithmetic; if they curve, think geometric.

Section 4

The Four Formulas You Need

Every sequence in this lesson can be described by a recursive formula (step-by-step) or an explicit formula (direct jump). Let's break down each one, starting with arithmetic sequences and then moving to geometric ones.

Arithmetic Sequences

Arithmetic — Recursive Formula

a₁ = (first term), aₙ = aₙ₋₁ + d

aₙ₋₁ = the previous term | d = common difference (what you add each time)

The recursive formula says: "start at a₁, then to find any term, take the one before it and add d." It's simple and intuitive, but if you want the 50th term, you'd have to calculate all 49 terms before it. That's where the explicit formula helps.

Arithmetic — Explicit Formula

aₙ = a₁ + (n − 1) × d

aₙ = the nth term | a₁ = first term | n = position number | d = common difference

Why (n − 1) and not just n? Because when n = 1 (the very first term), you haven't added d yet — you've added it zero times. By the second term you've added d once; by the third, twice; and so on. So you always add d one fewer time than the term number.

Geometric Sequences

Geometric — Recursive Formula

a₁ = (first term), aₙ = aₙ₋₁ × r

aₙ₋₁ = the previous term | r = common ratio (what you multiply by each time)

Same idea as before, but instead of adding a constant, you're multiplying by one. If r = 3, each term is triple the one before it.

Geometric — Explicit Formula

aₙ = a₁ × r^(n−1)

aₙ = the nth term | a₁ = first term | r = common ratio | n − 1 = number of times you multiply

Again, the exponent is (n − 1) because at the first term you haven't multiplied by r at all — r⁰ = 1, which gives you back a₁ itself. Notice how the arithmetic formula uses addition while the geometric formula uses multiplication (exponent). That's the fundamental difference.

✦ Key Takeaway

Think of the relationship like this: addition is to arithmetic sequences as multiplication is to geometric sequences. The recursive formula is your turn-by-turn GPS directions. The explicit formula is the street address — it gets you exactly where you need to go, no matter how far away, in a single step.

Section 5

Identifying & Classifying Sequences

When you're given a sequence of numbers and asked to write a formula, the first step is to identify what type it is. Here's a systematic approach: check the differences between consecutive terms first; if they're all the same, it's arithmetic. If not, check the ratios; if those are all the same, it's geometric. The flowchart below walks you through this process.

Let's apply this to a quick example. Given the sequence 5, 15, 45, 135, …, compute the differences: 15 − 5 = 10, 45 − 15 = 30, 135 − 45 = 90. The differences (10, 30, 90) are not equal, so it's not arithmetic. Now try ratios: 15 ÷ 5 = 3, 45 ÷ 15 = 3, 135 ÷ 45 = 3. Constant ratio of r = 3 — it's geometric!

Side-by-Side Reference

Feature	Arithmetic Sequence	Geometric Sequence
Growth rule	Add the common difference `d`	Multiply by the common ratio `r`
Recursive formula	`aₙ = aₙ₋₁ + d`	`aₙ = aₙ₋₁ × r`
Explicit formula	`aₙ = a₁ + (n − 1)d`	aₙ = a₁ × rⁿ⁻¹
Graph shape	Straight line (linear)	Curve (exponential)
Related function	Linear: `f(x) = mx + b`	Exponential: f(x) = a × b^x
Real-world example	Saving $50 every month	Bacteria doubling every hour

Section 6

Worked Example: Modeling a Real Situation

Let's put everything together with a realistic problem that walks through every step from identifying the sequence type to writing both formulas and using them to make predictions.

Coffee Shop Monthly Earnings

ProblemA new coffee shop earns $800 in its first month. Each month after that, it earns $150 more than the previous month as it gains regular customers. Write a recursive formula and an explicit formula for the monthly earnings. Then predict the shop's earnings in month 12.

Step 1 — Identify the TypeThe earnings increase by a constant amount of $150 each month. That means this is an arithmetic sequence with a₁ = 800 and d = 150.

Step 2 — Write the Recursive FormulaState the first term and the rule for getting the next one:

a₁ = 800, aₙ = aₙ₋₁ + 150 — This tells us: "Start at 800. To find any month's earnings, take the previous month's earnings and add 150."

Step 3 — Write the Explicit FormulaUse the arithmetic explicit formula aₙ = a₁ + (n − 1) × d and substitute:

aₙ = 800 + (n − 1) × 150 → Simplify by distributing: aₙ = 800 + 150n − 150 = 650 + 150n

Step 4 — Find the Earnings in Month 12Plug n = 12 into the explicit formula:

a₁₂ = 650 + 150(12) = 650 + 1800 = $2,450

Step 5 — Interpret the ResultThe coffee shop is predicted to earn $2,450 in its 12th month. Notice how much faster this was than calculating all 12 terms one by one — that's the power of the explicit formula. Also note that the simplified explicit formula aₙ = 650 + 150n looks exactly like a linear equation y = mx + b with slope 150 and y-intercept 650, which confirms the arithmetic/linear connection we saw in the graph.

Section 7

Recursive vs. Explicit — Strengths & Limitations

Both formula types describe the same sequence, but each has situations where it shines and situations where it struggles. Knowing when to use each one is an important part of mathematical modeling.

Criteria	Recursive Formula	Explicit Formula
Ease of writing	Very easy — just state "add `d`" or "multiply by `r`"	Requires knowing the pattern for the formula structure
Finding the next term	✅ Ideal — one quick operation	Works, but unnecessary extra setup
Finding the 100th term	❌ Must compute all 99 terms before it	✅ Plug in `n = 100` directly
Spreadsheet use	✅ Natural fit — each cell references the one above it	Also works — each cell uses its row number
Graphing	Harder to graph without generating many points first	✅ Directly translates to a function you can graph
Real-world modeling	Great when you know month-to-month changes	Great when you need a prediction far into the future

✦ Key Takeaway

Think of recursive and explicit formulas as two tools in the same toolbox. A recursive formula is like a recipe that says "add one more cup of flour each time" — perfect when you're cooking step by step. An explicit formula is like a nutrition label that tells you the total flour for any batch size — perfect when you want a quick answer without doing every step. Smart problem-solvers learn to switch between them depending on what the question asks.

Section 8

Connection to Functions & Beyond

You may have noticed that arithmetic sequences look a lot like linear functions and geometric sequences look like exponential functions. That's not a coincidence — sequences are actually a special case of functions where the input (n) must be a positive integer (1, 2, 3, …) rather than any real number.

Concept This Lesson	Advanced Version (Future Courses)
Arithmetic sequence: `aₙ = a₁ + (n−1)d`	Linear function: `f(x) = mx + b`
Geometric sequence: aₙ = a₁ × rⁿ⁻¹	Exponential function: f(x) = a × b^x
Common difference `d`	Slope `m` (rate of change)
Common ratio `r`	Base `b` (growth/decay factor)
Summing a finite number of terms	Series & sigma notation (Algebra 2, Precalculus)
Recursive definition: `aₙ = f(aₙ₋₁)`	Recurrence relations (Discrete Math, Computer Science)

In Algebra 2 and Precalculus, you'll learn how to sum up all the terms in a sequence (called a series), which has powerful applications in finance (calculating loan payments), physics (adding up forces), and computer science (analyzing algorithm efficiency). The recursive thinking you practice here also forms the basis of recursion in programming, where a function calls itself to solve smaller and smaller pieces of a problem. So the skills you're building now are the foundation for much more advanced work later.

Section 9

Practice Problems

Test your understanding with these five problems. They start with a conceptual check and build up to a real-world modeling challenge. Try each one before revealing the answer.

PROBLEM 1 — CONCEPTUAL

A student claims that the sequence 4, 8, 16, 32, … is arithmetic because "it keeps going up." Explain why the student is wrong, and identify what type of sequence it actually is.

PROBLEM 2 — BASIC CALCULATION

For the arithmetic sequence 7, 12, 17, 22, …, write both a recursive formula and an explicit formula. Then find a₁₀ (the 10th term).

PROBLEM 3 — INTERMEDIATE

A geometric sequence has a₁ = 3 and a₄ = 375. Find the common ratio r, write the explicit formula, and determine a₆.

PROBLEM 4 — APPLIED / MULTI-STEP

A town's population is 12,000 and it grows by 4% each year. Write an explicit formula modeling the population after n years. What will the population be after 8 years? (Round to the nearest whole number.) Is this arithmetic or geometric, and why?

PROBLEM 5 — CRITICAL THINKING

Maya starts a savings plan where she deposits $200 in month 1 and increases her deposit by $25 each month. Ethan also saves, but he deposits $200 in month 1 and increases his deposit by 10% each month. Write explicit formulas for both. Who deposits more in month 10? In month 20? What does this tell you about the long-term behavior of arithmetic vs. geometric growth?

Summary

Lesson Summary

In this lesson you learned the two most fundamental types of sequences. An arithmetic sequence grows by adding a constant called the common difference (d), producing a linear pattern, while a geometric sequence grows by multiplying by a constant called the common ratio (r), producing an exponential curve. For each type, you can write a recursive formula — which defines each term based on the one before it — or an explicit formula — which calculates any term directly from its position number n.

The four key formulas are: aₙ = aₙ₋₁ + d (arithmetic recursive), aₙ = a₁ + (n − 1)d (arithmetic explicit), aₙ = aₙ₋₁ × r (geometric recursive), and aₙ = a₁ × rⁿ⁻¹ (geometric explicit). To identify a sequence's type, check whether the differences or the ratios between consecutive terms are constant. These formulas let you model a wide range of real situations — from savings accounts and salary increases to population growth and depreciation — giving you the power to predict values far into the future without tedious term-by-term calculation.