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## Algebra 1

### Course: Algebra 1>Unit 9

Lesson 1: Introduction to arithmetic sequences

# Intro to arithmetic sequences

Get comfortable with sequences in general, and learn what arithmetic sequences are.
Before you take this lesson, make sure you know how to add and subtract negative numbers.

## What is a sequence?

Here are a few lists of numbers:
• 3, 5, 7 ...
• 21, 16, 11, 6 ...
• 1, 2, 4, 8 ...
Ordered lists of numbers like these are called sequences. Each number in a sequence is called a term.
$3,$$5,$$7,\text{…}$
$↑$$↑$$↑$
Sequences usually have patterns that allow us to predict what the next term might be.
For example, in the sequence 3, 5, 7 ..., you always add two to get the next term:
$+2\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$
$3,$$5,$$7,\text{…}$
The three dots that come at the end indicate that the sequence can be extended, even though we only see a few terms.
We can do so by using the pattern.
For example, the fourth term of the sequence should be nine, the fifth term should be 11, etc.
$+2\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$
$3,$$5,$$7,$$9,$$11,\text{…}$

Extend the sequences according to their pattern.
Problem 1
Pattern: Add five to the previous term.
$+5\phantom{\rule{0.167em}{0ex}}↷$$+5\phantom{\rule{0.167em}{0ex}}↷$$+5\phantom{\rule{0.167em}{0ex}}↷$
$3,$$8,$$13,$
$,\text{…}$

Problem 2
Pattern: Subtract three from the previous term.
$-3\phantom{\rule{0.167em}{0ex}}↷$$-3\phantom{\rule{0.167em}{0ex}}↷$$-3\phantom{\rule{0.167em}{0ex}}↷$
$20,$$17,$$14,$
$,\text{…}$

Problem 3
Pattern: Multiply the previous term by two.
$×2\phantom{\rule{0.167em}{0ex}}↷$$×2\phantom{\rule{0.167em}{0ex}}↷$$×2\phantom{\rule{0.167em}{0ex}}↷$
$3,$$6,$$12,$
$,\text{…}$

Problem 4
Match each sequence with its pattern.

## What is an arithmetic sequence?

For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called arithmetic sequences.
In an arithmetic sequence, the difference between consecutive terms is always the same.
For example, the sequence 3, 5, 7, 9 ... is arithmetic because the difference between consecutive terms is always two.
$+2\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$
$3,$$5,$$7,$$9,\text{…}$
The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always minus five.
$-5\phantom{\rule{0.167em}{0ex}}↷$$-5\phantom{\rule{0.167em}{0ex}}↷$$-5\phantom{\rule{0.167em}{0ex}}↷$
$21,$$16,$$11,$$6,\text{…}$
The sequence 1, 2, 4, 8 ... is not arithmetic because the difference between consecutive terms is not the same.
$+1\phantom{\rule{0.167em}{0ex}}↷$$+2\phantom{\rule{0.167em}{0ex}}↷$$+4\phantom{\rule{0.167em}{0ex}}↷$
$1,$$2,$$4,$$8,\text{…}$

Problem 5
Select all arithmetic sequences.

Problem 6
The first term of a sequence is one. Which of the following patterns would make the sequence arithmetic?

## The common difference

The common difference of an arithmetic sequence is the constant difference between consecutive terms.
For example, the common difference of 10, 21, 32, 43 ... is 11:
$+11\phantom{\rule{0.167em}{0ex}}↷$$+11\phantom{\rule{0.167em}{0ex}}↷$$+11\phantom{\rule{0.167em}{0ex}}↷$
$10,$$21,$$32,$$43,\text{…}$
The common difference of –2, –5, –8, –11 ... is negative three:
$-3\phantom{\rule{0.167em}{0ex}}↷$$-3\phantom{\rule{0.167em}{0ex}}↷$$-3\phantom{\rule{0.167em}{0ex}}↷$
$-2,$$-5,$$-8,$$-11,\text{…}$

Problem 7
What is the common difference of 2, 8, 14, 20 ...?

Problem 8
What is the common difference of $5,2,-1,-4\text{…}$?

Problem 9
What is the common difference of $1,\phantom{\rule{0.167em}{0ex}}1\frac{1}{3},1\frac{2}{3},\phantom{\rule{0.167em}{0ex}}2,\text{…}$?

Reflection question
What must be true about an arithmetic sequence whose common difference is negative?

Challenge problem
The first term of an arithmetic sequence is 10 and its common difference is negative seven.
What is the fourth term of the sequence?

## What's next?

Learn about formulas of arithmetic sequences, which give us the information we need to find any term in the sequence.

## Want to join the conversation?

• is the lucas series series also an arithmetic sequence
eg. {1,1,2,3,5,8,13,21,34...} where asub(k)=asub(k-1)+asub(k-2)
• NO. Take a look at the difference between the terms of the sequence. The difference between the terms is not constant (not the same), hence not an arithmetic sequence.
• So if adding and subtracting from the previous terms create an arithmetic sequence, would multiplying or dividing make a geometric sequence?
• In short, yes.

Arithmetic is always adding or subtracting the same constant term or amount.
Geometric is always multiplying or dividing by the same constant amount.
• Are arithmetic sequences always either addition or subtraction
• Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.
• this is so easy but Sal somehow made it look like it's some kind of a more complicated thing
• Instead of learning it in the book, my teacher says to learn on here but its hard when I'm a visual learner XD
• do all arithmetic sequences have to have real numbers?
• A sequence can be of unreal numbers I think that arithmetic progression should of real numbers
• can somebody please explain why we cannot multiply or divide sequentially and still have the sequence be arithmetic?
• Some historical mathematician defined arithmetic sequences has being defined by addition/subtractions of a common value to get from one term to the next.

Geometric sequences were defined as using multiplication/division by a common value to get from one term to the next.