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.

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.

Are arithmetic sequences always either addition or subtraction

Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.

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.

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

If I multiple the last number by a fixed number in a sequence,is it not that an arithmetic sequences

Simply put if its multiplied or divided it'll be geometric whereas if its added or subtracted its an arithmetic secuence

I put that 3, 9, 27, and 81 was a arithmetic sequence. Why is that wrong? I know why but I just feel like that should be Arithmetic.

In this case that is a geometric sequence in which you are multiplying or dividing by a number. An arithmetic sequence would be where you are adding or subtracting by the same number. In this case 3,9,27... is not arithemtic.

Why does multiplying or dividing count as an arithmetic sequence?

They don't. An *arithmetic sequence* uses addition/subtraction of a common value to create the next term in the sequence. A *geometric sequences* uses multiplication/division of a common value to create the next term in the sequence. Hope this helps.

Main content

Course: Algebra 1 (Eureka Math/EngageNY) > Unit 3

Lesson 1: Topic A: Lessons 1-3: Arithmetic sequence intro

Intro to arithmetic sequences

Google Classroom

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.

$3,$ ‍	$5,$ ‍	$7, …$ ‍
$↑$ ‍	$↑$ ‍	$↑$ ‍
$1^{st} term$ ‍	$2^{nd} term$ ‍	$3^{rd} term$ ‍

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:

	$+ 2 ↷$ ‍		$+ 2 ↷$ ‍
$3,$ ‍		$5,$ ‍		$7, …$ ‍

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.

	$+ 2 ↷$ ‍		$+ 2 ↷$ ‍		$+ 2 ↷$ ‍		$+ 2 ↷$ ‍
$3,$ ‍		$5,$ ‍		$7,$ ‍		$9,$ ‍		$11, …$ ‍

Check your understanding

Extend the sequences according to their pattern.

Problem 1

Pattern: Add five to the previous term.

	$+ 5 ↷$ ‍		$+ 5 ↷$ ‍		$+ 5 ↷$ ‍
$3,$ ‍		$8,$ ‍		$13,$ ‍		Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $, …$ ‍

Problem 2

Pattern: Subtract three from the previous term.

	$- 3 ↷$ ‍		$- 3 ↷$ ‍		$- 3 ↷$ ‍
$20,$ ‍		$17,$ ‍		$14,$ ‍		Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $, …$ ‍

Problem 3

Pattern: Multiply the previous term by two.

	$\times 2 ↷$ ‍		$\times 2 ↷$ ‍		$\times 2 ↷$ ‍
$3,$ ‍		$6,$ ‍		$12,$ ‍		Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $, …$ ‍

	$+ 5 ↷$ ‍		$+ 5 ↷$ ‍		$+ 5 ↷$ ‍
$3,$ ‍		$8,$ ‍		$13,$ ‍		$18, …$ ‍

	$- 3 ↷$ ‍		$- 3 ↷$ ‍		$- 3 ↷$ ‍
$20,$ ‍		$17,$ ‍		$14,$ ‍		$11, …$ ‍

	$\times 2 ↷$ ‍		$\times 2 ↷$ ‍		$\times 2 ↷$ ‍
$3,$ ‍		$6,$ ‍		$12,$ ‍		$24, …$ ‍

Problem 4

Match each sequence with its pattern.

1

Here are the sequences with their patterns indicated:

	$+ 7 ↷$ ‍		$+ 7 ↷$ ‍		$+ 7 ↷$ ‍
$4,$ ‍		$11,$ ‍		$18,$ ‍		$25, …$ ‍

	$\div 2 ↷$ ‍		$\div 2 ↷$ ‍		$\div 2 ↷$ ‍
$40,$ ‍		$20,$ ‍		$10,$ ‍		$5, …$ ‍

	$- 4 ↷$ ‍		$- 4 ↷$ ‍		$- 4 ↷$ ‍
$100,$ ‍		$96,$ ‍		$92,$ ‍		$88, …$ ‍

	$\times 3 ↷$ ‍		$\times 3 ↷$ ‍		$\times 3 ↷$ ‍
$4,$ ‍		$12,$ ‍		$36,$ ‍		$108, …$ ‍

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.

	$+ 2 ↷$ ‍		$+ 2 ↷$ ‍		$+ 2 ↷$ ‍
$3,$ ‍		$5,$ ‍		$7,$ ‍		$9, …$ ‍

The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always minus five.

	$- 5 ↷$ ‍		$- 5 ↷$ ‍		$- 5 ↷$ ‍
$21,$ ‍		$16,$ ‍		$11,$ ‍		$6, …$ ‍

The sequence 1, 2, 4, 8 ... is not arithmetic because the difference between consecutive terms is not the same.

	$+ 1 ↷$ ‍		$+ 2 ↷$ ‍		$+ 4 ↷$ ‍
$1,$ ‍		$2,$ ‍		$4,$ ‍		$8, …$ ‍

Check your understanding

Problem 5

Select all arithmetic sequences.

Here are the sequences with their consecutive differences:

	$+ 4 ↷$ ‍		$+ 4 ↷$ ‍		$+ 4 ↷$ ‍
$11,$ ‍		$15,$ ‍		$19,$ ‍		$23, …$ ‍

The difference between consecutive terms in the sequence above is always four, so this is an arithmetic sequence.

	$+ 1 ↷$ ‍		$+ 2 ↷$ ‍		$+ 3 ↷$ ‍
$0,$ ‍		$1,$ ‍		$3,$ ‍		$6, …$ ‍

The difference between consecutive terms in the sequence above is changing, so this is not an arithmetic sequence.

	$+ 6 ↷$ ‍		$+ 18 ↷$ ‍		$+ 54 ↷$ ‍
$3,$ ‍		$9,$ ‍		$27,$ ‍		$81, …$ ‍

The difference between consecutive terms in the sequence above is changing, so this is not an arithmetic sequence.

	$- 9 ↷$ ‍		$- 9 ↷$ ‍		$- 9 ↷$ ‍
$29,$ ‍		$20,$ ‍		$11,$ ‍		$2, …$ ‍

The difference between consecutive terms in the sequence above is always negative nine, so this is an arithmetic sequence.

Problem 6

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

The meaning of the pattern "add four to the previous term" is that the difference between consecutive terms is always four. So, this pattern describes an arithmetic sequence.

	$+ 4 ↷$ ‍		$+ 4 ↷$ ‍		$+ 4 ↷$ ‍
$1,$ ‍		$5,$ ‍		$9,$ ‍		$13, …$ ‍

In a similar manner, the pattern "subtract four from the previous term" describes an arithmetic sequence where the difference between consecutive terms is always negative four.

	$- 4 ↷$ ‍		$- 4 ↷$ ‍		$- 4 ↷$ ‍
$1,$ ‍		$- 3,$ ‍		$- 7,$ ‍		$- 11, …$ ‍

In contrast, the pattern "multiply the previous term by four" doesn't result in a sequence where the difference between consecutive terms is constant.

	$+ 3 ↷$ ‍		$+ 12 ↷$ ‍		$+ 48 ↷$ ‍
$1,$ ‍		$4,$ ‍		$16,$ ‍		$64, …$ ‍

In a similar manner, the pattern "divide the previous term by four" doesn't result in an arithmetic sequence.

	$- \frac{3}{4} ↷$ ‍		$- \frac{3}{16} ↷$ ‍		$- \frac{3}{64} ↷$ ‍
$1,$ ‍		$\frac{1}{4},$ ‍		$\frac{1}{16},$ ‍		$\frac{1}{64}, …$ ‍

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:

	$+ 11 ↷$ ‍		$+ 11 ↷$ ‍		$+ 11 ↷$ ‍
$10,$ ‍		$21,$ ‍		$32,$ ‍		$43, …$ ‍

The common difference of –2, –5, –8, –11 ... is negative three:

	$- 3 ↷$ ‍		$- 3 ↷$ ‍		$- 3 ↷$ ‍
$- 2,$ ‍		$- 5,$ ‍		$- 8,$ ‍		$- 11, …$ ‍

Check your understanding

Problem 7

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

	$+ 6 ↷$ ‍		$+ 6 ↷$ ‍		$+ 6 ↷$ ‍
$2,$ ‍		$8,$ ‍		$14,$ ‍		$20, …$ ‍

Problem 8

What is the common difference of $5, 2, - 1, - 4 …$ ‍?

	$- 3 ↷$ ‍		$- 3 ↷$ ‍		$- 3 ↷$ ‍
$5,$ ‍		$2,$ ‍		$- 1,$ ‍		$- 4, …$ ‍

Problem 9

What is the common difference of $1, 1 \frac{1}{3}, 1 \frac{2}{3}, 2, …$ ‍?

	$+ \frac{1}{3} ↷$ ‍		$+ \frac{1}{3} ↷$ ‍		$+ \frac{1}{3} ↷$ ‍
$1$ ‍		$1 \frac{1}{3},$ ‍		$1 \frac{2}{3},$ ‍		$2, …$ ‍

Reflection question

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

A negative common difference means we need to subtract something from any term to get to the next term. In other words, the terms are always decreasing.

It isn't necessary that all the terms in the sequence are positive or that all of them are negative. Here are arithmetic sequences that have both negative and positive terms, but they all have negative common differences.

	$- 1 ↷$ ‍		$- 1 ↷$ ‍		$- 1 ↷$ ‍
$2,$ ‍		$1,$ ‍		$0,$ ‍		$- 1, …$ ‍

	$- 5 ↷$ ‍		$- 5 ↷$ ‍		$- 5 ↷$ ‍
$7,$ ‍		$2,$ ‍		$- 3,$ ‍		$- 8, …$ ‍

	$- 8 ↷$ ‍		$- 8 ↷$ ‍		$- 8 ↷$ ‍
$3,$ ‍		$- 5,$ ‍		$- 13,$ ‍		$- 21, …$ ‍

Note how all of the above sequences have terms that are always decreasing.

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?

We are given the first term, 10, and the common difference, negative seven. With this information, we can extend the sequence until we reach the fourth term:

	$- 7 ↷$ ‍		$- 7 ↷$ ‍		$- 7 ↷$ ‍
$10,$ ‍		$3,$ ‍		$- 4,$ ‍		$- 11, …$ ‍

The fourth term of the sequence is -11.

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?

Sort by:

shivansh chauhan
Posted 8 years ago. Direct link to shivansh chauhan's post “is the lucas series seri...”
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)
Button navigates to signup pageComment on shivansh chauhan's post “is the lucas series seri...”
(19 votes)
Answer
- David Schwartz
  Posted 4 years ago. Direct link to David Schwartz's post “NO. Take a look at the di...”
  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.
  Button navigates to signup page
  (53 votes)
Anne Joseph
Posted 7 years ago. Direct link to Anne Joseph's post “So if adding and subtract...”
So if adding and subtracting from the previous terms create an arithmetic sequence, would multiplying or dividing make a geometric sequence?
Button navigates to signup pageComment on Anne Joseph's post “So if adding and subtract...”
(27 votes)
Answer
- Matt Ahlberg
  Posted 7 years ago. Direct link to Matt Ahlberg's post “In short, yes. Arithmeti...”
  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.
  Button navigates to signup page
  (44 votes)
Kat Tracy
Posted 6 years ago. Direct link to Kat Tracy's post “Are arithmetic sequences ...”
Are arithmetic sequences always either addition or subtraction
Button navigates to signup pageButton navigates to signup page
(17 votes)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “Yes that is what makes th...”
  Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.
  Button navigates to signup page
  (28 votes)
sarra
Posted 9 months ago. Direct link to sarra's post “this is so easy but Sal s...”
this is so easy but Sal somehow made it look like it's some kind of a more complicated thing
Button navigates to signup pageComment on sarra's post “this is so easy but Sal s...”
(13 votes)
Answer
Shepherd.E
Posted 5 years ago. Direct link to Shepherd.E's post “Instead of learning it in...”
Instead of learning it in the book, my teacher says to learn on here but its hard when I'm a visual learner XD
Button navigates to signup pageButton navigates to signup page
(10 votes)
Answer
- Cleve
  Posted a month ago. Direct link to Cleve's post “I thought it would be eas...”
  I thought it would be easier here with the videos
  Button navigates to signup page
  (1 vote)
dylan.forr99
Posted a year ago. Direct link to dylan.forr99's post “can somebody please expla...”
can somebody please explain why we cannot multiply or divide sequentially and still have the sequence be arithmetic?
Button navigates to signup pageComment on dylan.forr99's post “can somebody please expla...”
(5 votes)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “Some historical mathemati...”
  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.
  Comment on Kim Seidel's post “Some historical mathemati...”
  (13 votes)
Julian C. Gonzales
Posted 7 years ago. Direct link to Julian C. Gonzales's post “do all arithmetic sequenc...”
do all arithmetic sequences have to have real numbers?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Safia Khan
  Posted 6 years ago. Direct link to Safia Khan's post “A sequence can be of unre...”
  A sequence can be of unreal numbers I think that arithmetic progression should of real numbers
  Button navigates to signup page
  (11 votes)
Unenthusiastic Learner
Posted 8 months ago. Direct link to Unenthusiastic Learner's post “I put that 3, 9, 27, and ...”
I put that 3, 9, 27, and 81 was a arithmetic sequence. Why is that wrong? I know why but I just feel like that should be Arithmetic.
Button navigates to signup pageComment on Unenthusiastic Learner's post “I put that 3, 9, 27, and ...”
(4 votes)
Answer
- j.shin.scott
  Posted 8 months ago. Direct link to j.shin.scott's post “In this case that is a ge...”
  In this case that is a geometric sequence in which you are multiplying or dividing by a number. An arithmetic sequence would be where you are adding or subtracting by the same number. In this case 3,9,27... is not arithemtic.
  Button navigates to signup page
  (10 votes)
maria kautondokwa
Posted 5 years ago. Direct link to maria kautondokwa's post “If I multiple the last nu...”
If I multiple the last number by a fixed number in a sequence,is it not that an arithmetic sequences
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- WilliamBenSchool
  Posted a year ago. Direct link to WilliamBenSchool's post “Simply put if its multipl...”
  Simply put if its multiplied or divided it'll be geometric whereas if its added or subtracted its an arithmetic secuence
  Button navigates to signup page
  (2 votes)
Timothy
Posted a year ago. Direct link to Timothy's post “Why does multiplying or d...”
Why does multiplying or dividing count as an arithmetic sequence?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “They don't. An *arithmet...”
  They don't.
  
  An arithmetic sequence uses addition/subtraction of a common value to create the next term in the sequence.
  
  A geometric sequences uses multiplication/division of a common value to create the next term in the sequence.
  
  Hope this helps.
  Comment on Kim Seidel's post “They don't. An *arithmet...”
  (15 votes)