Algebra Mastery

The Human Knowledge Project

Chapter 17 — Sequences and Series

17.1 Learning Objectives

By the end of this chapter, you should be able to:

understand what sequences are

distinguish sequences from functions

identify arithmetic sequences

identify geometric sequences

find explicit formulas for sequences

find recursive formulas

evaluate terms in sequences

understand finite and infinite series

compute arithmetic and geometric sums

connect sequences to real-world growth systems

17.2 Big Picture — Sequences Describe Ordered Patterns

A sequence is:

an ordered list of numbers

Examples:

1, 2, 3, 4, 5

2, 4, 8, 16, 32

Sequences appear everywhere:

finance

population growth

computing

engineering

signal processing

physics

economics

Sequences help humans understand:

patterns

growth

repetition

prediction

long-term behavior

17.3 What Is a Sequence?

A sequence assigns:

a value

to each position number

Example:

2, 5, 8, 11, 14

Terms are ordered carefully.

Each term has:

a position

an index

17.4 Sequence Notation

Notation:

a₁, a₂, a₃, ...

means:

first term

second term

third term

Example:

a₁ = 3

a₂ = 6

a₃ = 9

17.5 Arithmetic Sequences

Arithmetic sequences grow by:

repeated addition

Example:

3, 7, 11, 15, 19

Each step adds:

This constant difference is called:

common difference

17.6 Explicit Formula for Arithmetic Sequences

Formula:

a_n = a₁ + (n - 1)d

Where:

a₁ = first term

d = common difference

n = position

17.7 Arithmetic Example

Sequence:

5, 8, 11, 14

Here:

first term = 5

difference = 3

Formula:

a_n = 5 + (n - 1)(3)

17.8 Geometric Sequences

Geometric sequences grow by:

repeated multiplication

Example:

2, 6, 18, 54

Each term multiplies by:

This multiplier is called:

common ratio

17.9 Explicit Formula for Geometric Sequences

Formula:

a_n = a₁r^(n-1)

Where:

a₁ = first term

r = common ratio

17.10 Geometric Example

Sequence:

3, 6, 12, 24

First term:

Ratio:

Formula:

a_n = 3(2)^(n-1)

17.11 Recursive Sequences

Recursive formulas define terms using previous terms.

Example:

a_n = a_(n-1) + 2

starting with:

a₁ = 1

This creates:

1, 3, 5, 7, 9

17.12 Finite vs Infinite Sequences

Finite sequences:

end

Example:

1, 2, 3, 4

Infinite sequences:

continue forever

Example:

1, 2, 3, 4, ...

17.13 Series

A series is:

the SUM of sequence terms

Example:

1 + 2 + 3 + 4

Series analyze:

accumulation

total growth

combined effects

17.14 Arithmetic Series

Example:

2 + 4 + 6 + 8

Formula:

S_n = n(a₁ + a_n)/2

Where:

n = number of terms

17.15 Arithmetic Series Example

Find:

1 + 2 + 3 + ... + 10

Use formula:

S = 10(1 + 10)/2

Result:

17.16 Geometric Series

Example:

2 + 4 + 8 + 16

Formula:

S_n = a₁(1 - rⁿ)/(1 - r)

provided:

r ≠ 1

17.17 Infinite Geometric Series (Intro)

Some infinite geometric series converge.

Example:

1 + 1/2 + 1/4 + 1/8 + ...

approaches:

This idea becomes extremely important in calculus.

17.18 Real-World Sequences

Examples:

compound interest

population growth

computer memory

AI learning cycles

network scaling

radioactive decay

signal processing

Sequences model repeated processes over time.

17.19 Common Beginner Difficulties

Students often struggle with:

arithmetic vs geometric distinction

recursive formulas

indexing errors

formula substitution

series summation

exponential growth intuition

These struggles are normal.

Sequence fluency develops through:

repetition

pattern recognition

careful organization

experimentation

17.20 Mental Model

Sequences are:

structured patterns through time

Series are:

accumulated totals of those patterns

They help humans model:

growth

repetition

accumulation

prediction

17.21 Warm-Up Problems

Problems

Identify whether sequence is:

arithmetic

geometric

2, 4, 6, 8

Identify sequence type:

3, 6, 12, 24

Find next term:

5, 8, 11, 14

Find next term:

2, 6, 18, 54

Find common difference:

4, 9, 14, 19

Find common ratio:

2, 10, 50, 250

Evaluate:

a_n = 2n + 1

a_4

Evaluate:

a_n = 3(2)^(n-1)

a_3

Find sum:

1 + 2 + 3 + 4 + 5

Find sum:

2 + 4 + 6 + 8

Identify whether finite or infinite:

1, 2, 3, ...

Identify whether finite or infinite:

4, 8, 12

17.22 Guided Problems

Problems

Find explicit arithmetic formula:

3, 7, 11, 15

Find explicit geometric formula:

2, 6, 18, 54

Find the 10th arithmetic term:

2, 5, 8, 11

Find the 6th geometric term:

3, 6, 12, 24

Find arithmetic series sum:

1 + 2 + ... + 20

Find geometric series sum:

2 + 4 + 8 + 16

Write recursive formula:

5, 8, 11, 14

Write recursive formula:

2, 4, 8, 16

Explain difference between:

arithmetic

geometric sequences

Explain why geometric growth becomes large quickly.

Describe a real-world sequence.

Explain why series are useful.

17.23 Challenge Problems

Find the 12th arithmetic term:

4, 9, 14, 19

Find the 7th geometric term:

2, 6, 18, 54

Find arithmetic series sum:

5 + 10 + 15 + ... + 50

Find geometric series sum:

1 + 3 + 9 + 27

Determine sequence type:

1, 4, 9, 16

Determine sequence type:

100, 50, 25, 12.5

Explain why arithmetic sequences grow steadily.

Explain why geometric sequences accelerate.

Describe a scientific or computing system involving sequences.

Explain why sequences and series are important in higher mathematics.

17.24 Solutions

Solutions to Warm-Up Problems

arithmetic

geometric

162

infinite

finite

Solutions to Guided Problems

a_n = 3 + (n - 1)(4)

a_n = 2(3)^(n-1)

210

a_n = a_(n-1) + 3

a_n = 2a_(n-1)

Arithmetic sequences add repeatedly.

Geometric sequences multiply repeatedly.

Repeated multiplication compounds rapidly.

Examples include:

population growth

finance

computer memory scaling

radioactive decay

Series model accumulated totals and long-term growth.

Solutions to Challenge Problems

1458

275

neither arithmetic nor geometric

geometric

Arithmetic sequences change by constant addition.

Geometric sequences compound through repeated multiplication.

Examples include:

AI learning cycles

computer networking

finance

signal processing

population systems

Sequences and series provide foundational tools for modeling growth, accumulation, recursion, and long-term behavior throughout mathematics and science.