Main content
Algebra 1
Course: Algebra 1 > Unit 9
Lesson 1: Introduction to arithmetic sequences- Sequences intro
- Intro to arithmetic sequences
- Intro to arithmetic sequences
- Extending arithmetic sequences
- Extend arithmetic sequences
- Using arithmetic sequences formulas
- Intro to arithmetic sequence formulas
- Worked example: using recursive formula for arithmetic sequence
- Use arithmetic sequence formulas
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice
Intro to arithmetic sequences
CCSS.Math:
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, comma | 5, comma | 7, comma, point, point, point |
---|---|---|
\uparrow | \uparrow | \uparrow |
1, start superscript, start text, s, t, end text, end superscript, start text, space, t, e, r, m, end text | 2, start superscript, start text, n, d, end text, end superscript, start text, space, t, e, r, m, end text | 3, start superscript, start text, r, d, end text, end superscript, start text, space, t, e, r, m, end 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:
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | ||||
---|---|---|---|---|---|
3, comma | 5, comma | 7, comma, point, point, point |
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.
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | ||||||
---|---|---|---|---|---|---|---|---|---|
3, comma | 5, comma | 7, comma | 9, comma | 11, comma, point, point, point |
Check your understanding
Extend the sequences according to their 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.
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|
3, comma | 5, comma | 7, comma | 9, comma, point, point, point |
The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always minus five.
start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|
21, comma | 16, comma | 11, comma | 6, comma, point, point, point |
The sequence 1, 2, 4, 8 ... is not arithmetic because the difference between consecutive terms is not the same.
start color #ed5fa6, plus, 1, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 4, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|
1, comma | 2, comma | 4, comma | 8, comma, point, point, point |
Check your understanding
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:
start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|
10, comma | 21, comma | 32, comma | 43, comma, point, point, point |
The common difference of –2, –5, –8, –11 ... is negative three:
start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|
minus, 2, comma | minus, 5, comma | minus, 8, comma | minus, 11, comma, point, point, point |
Check your understanding
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)(19 votes)- 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.(42 votes)
- So if adding and subtracting from the previous terms create an arithmetic sequence, would multiplying or dividing make a geometric sequence?(23 votes)
- 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.(36 votes)
- Are arithmetic sequences always either addition or subtraction(13 votes)
- Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.(18 votes)
- Instead of learning it in the book, my teacher says to learn on here but its hard when I'm a visual learner XD(9 votes)
- do all arithmetic sequences have to have real numbers?(6 votes)
- A sequence can be of unreal numbers I think that arithmetic progression should of real numbers(11 votes)
- If I multiple the last number by a fixed number in a sequence,is it not that an arithmetic sequences(6 votes)
- Simply put if its multiplied or divided it'll be geometric whereas if its added or subtracted its an arithmetic secuence(2 votes)
- why I want to go home?(6 votes)
- cuz school sucks(2 votes)
- Is there many kinds of different formulas to write explicit and Recursive equations or there just one?(4 votes)
- There is basically one formula, you just have to change the numbers.(5 votes)
- i understand it but i was wondering if there is an easier
way to solve arithemic sequences(5 votes)- you'll see more about shortcuts when you get used to the formulas, the next few lessons explain them.(1 vote)
- can somebody please explain why we cannot multiply or divide sequentially and still have the sequence be arithmetic?(2 votes)
- 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.(6 votes)