Lesson 4.10

Arithmetic Sequences Intro

Numbers in a line, marching to a beat. If the step size is always the same, it's Arithmetic.

Introduction

A Sequence is just an ordered list of numbers. An Arithmetic Sequence is a special list where you add (or subtract) the same amount to get to the next number. We call this amount the Common Difference ().

Past Knowledge

Elementary School Patterns (Skip Counting). "2, 4, 6, 8..." is an arithmetic sequence.

Today's Goal

Identify if a sequence is arithmetic and find the common difference ().

Future Success

This connects directly to Linear Functions (). The difference is actually the Slope ().

Key Concepts

Finding "d"

To find the Common Difference (), take any term and subtract the previous term.

Sequence

5, 8, 11, 14...

8 - 5 = 3

11 - 8 = 3

d = 3

Sequence

20, 15, 10, 5...

15 - 20 = -5

10 - 15 = -5

d = -5

"Arithmetic" means adding. If numbers go down, you are adding a negative.

Worked Examples

Example 1: Continuing the Pattern

Basic

Find the next three terms:

Step 1: Find d

. . So .

Step 2: Add d repeatedly

23, 30, 37

Example 2: Is it Arithmetic?

Intermediate

Determine if is arithmetic.

Check Differences

Wait! The difference changed from 2 to 4.

NO, it is Geometric (multiplying by 2).

Example 3: Finding a specific term manually

Advanced

Find the 5th term () of sequence where and .

List them out

1st: 100

2nd: 90

3rd: 80

4th: 70

5th: 60

a_5 = 60

Common Pitfalls

Subtracting Backwards

To find , always do . Never do . If the sequence is going down, MUST be negative.

Thinking "Doubling" is Arithmetic

1, 2, 4, 8... feels like a pattern (and it is), but arithmetic specifically means adding.

Real-Life Applications

Stacking Cups: If 1 cup is 10cm tall, and each new cup adds 2cm to the stack height, finding the height of 50 cups is an arithmetic sequence problem (). You don't want to measure all 50.