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.

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

Is there many kinds of different formulas to write explicit and Recursive equations or there just one?

There is basically one formula, you just have to change the numbers.

As part of the questions above, I was asked to determine whether the sequence {3, 9, 27, 81...} was an arithmetic sequence. I answered yes. Yet, it told me that it was incorrect. If I multiply the previous term by 3, I should get the sequence stated. Could someone kindly shed some more light on that?

Yep. Arithmetic sequences are only addition or subtraction. (Ex. 1,2,3,4...) or (Ex. 10, 9, 8, 7...) Geometric sequences are when you multiply or divide. (Ex. 1, 10, 100, 1000...) or (Ex. 1000, 100, 10, 1) I hope this helps :)

Can anyone here explain in detail , what is the meaning of Recursive ?

Use any dictionary website to get the formal definition. With the recursive equation for a sequence, you must know the value of the prior term to create the next term. So, you follow a repetitive sequence of steps to get to the value you want. For example, to find the 4th term of a sequence using a recursive equation, you: 1) Calculate the 1st term (this is often given to you). 2) Use the value of the 1st term to calculate the 2nd term. 3) Use the value of the 2nd term to calculate the 3rd term. 4) Use the value of the 3rd term to calculate the 4th term. Basically, you can't get to the 4th term in one step. You have to go term to term up to the 4th term (or what ever term you want). If you want to get to the term directly, the explicit equation for a sequence would do that.

Main content

Algebra 1

Course: Algebra 1 > Unit 9

Lesson 1: Introduction to arithmetic sequences

Intro to arithmetic sequences

CCSS.Math:

HSF.IF.A.3

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, 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.

Problem 1

Pattern: Add five to the previous term.

	start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma		8, comma		13, comma		Your answer should be an integer, like 6 a simplified proper fraction, like 3, slash, 5 a simplified improper fraction, like 7, slash, 4 a mixed number, like 1, space, 3, slash, 4 an exact decimal, like 0, point, 75 a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text comma, point, point, point

Problem 2

Pattern: Subtract three from the previous term.

	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
20, comma		17, comma		14, comma		Your answer should be an integer, like 6 a simplified proper fraction, like 3, slash, 5 a simplified improper fraction, like 7, slash, 4 a mixed number, like 1, space, 3, slash, 4 an exact decimal, like 0, point, 75 a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text comma, point, point, point

Problem 3

Pattern: Multiply the previous term by two.

	start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
3, comma		6, comma		12, comma		Your answer should be an integer, like 6 a simplified proper fraction, like 3, slash, 5 a simplified improper fraction, like 7, slash, 4 a mixed number, like 1, space, 3, slash, 4 an exact decimal, like 0, point, 75 a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text comma, point, point, point

[I'm stuck! Please show me the solution.]

Problem 4

Match each sequence with its pattern.

[I need help!]

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

Problem 5

Select all arithmetic sequences.

(Choice A)
11, comma, 15, comma, 19, comma, 23, comma, point, point, point
(Choice B)
0, comma, 1, comma, 3, comma, 6, point, point, point, point
(Choice C)
3, comma, 9, comma, 27, comma, 81, comma, point, point, point
(Choice D)
29, comma, 20, comma, 11, comma, 2, comma, point, point, point

[I need help!]

Problem 6

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

[I need help!]

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

Problem 7

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

[I need help!]

Problem 8

What is the common difference of 5, comma, 2, comma, minus, 1, comma, minus, 4, point, point, point?

[I need help!]

Problem 9

What is the common difference of 1, comma, 1, start fraction, 1, divided by, 3, end fraction, comma, 1, start fraction, 2, divided by, 3, end fraction, comma, 2, comma, point, point, point?

[I need help!]

Reflection question

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

[I need help!]

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?

[I need help!]

What's next?

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

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Want to join the conversation?

shivansh chauhan
Posted 7 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 3 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
  (39 votes)
Anne Joseph
Posted 6 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 pageButton navigates to signup page
(22 votes)
Answer
- Matt Ahlberg
  Posted 6 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
  (33 votes)
Kat Tracy
Posted 5 years ago. Direct link to Kat Tracy's post “Are arithmetic sequences ...”
Are arithmetic sequences always either addition or subtraction
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(13 votes)
Answer
- David Severin
  Posted 5 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
  (16 votes)
Shepherd.E
Posted 4 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
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(8 votes)
Answer
Julian C. Gonzales
Posted 6 years ago. Direct link to Julian C. Gonzales's post “do all arithmetic sequenc...”
do all arithmetic sequences have to have real numbers?
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(6 votes)
Answer
- Safia Khan
  Posted 5 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
  (10 votes)
maria kautondokwa
Posted 4 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
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(6 votes)
Answer
- WilliamBenSchool
  Posted 4 months 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
  (1 vote)
stimulatingournervous
Posted 4 years ago. Direct link to stimulatingournervous's post “Is there many kinds of di...”
Is there many kinds of different formulas to write explicit and Recursive equations or there just one?
Button navigates to signup pageComment on stimulatingournervous's post “Is there many kinds of di...”
(4 votes)
Answer
- David Severin
  Posted 4 years ago. Direct link to David Severin's post “There is basically one fo...”
  There is basically one formula, you just have to change the numbers.
  Button navigates to signup page
  (5 votes)
Thaddeus Johnson
Posted 7 years ago. Direct link to Thaddeus Johnson's post “i understand it but i was...”
i understand it but i was wondering if there is an easier
way to solve arithemic sequences
Button navigates to signup pageComment on Thaddeus Johnson's post “i understand it but i was...”
(5 votes)
Answer
- 202ZBL
  Posted 2 years ago. Direct link to 202ZBL's post “you'll see more about sho...”
  you'll see more about shortcuts when you get used to the formulas, the next few lessons explain them.
  Button navigates to signup page
  (1 vote)
Success The Best💞
Posted 3 months ago. Direct link to Success The Best💞's post “As part of the questions ...”
As part of the questions above, I was asked to determine whether the sequence {3, 9, 27, 81...} was an arithmetic sequence. I answered yes. Yet, it told me that it was incorrect. If I multiply the previous term by 3, I should get the sequence stated. Could someone kindly shed some more light on that?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- MSTC_Grinding
  Posted 3 months ago. Direct link to MSTC_Grinding's post “Yep. Arithmetic sequences...”
  Yep. Arithmetic sequences are only addition or subtraction.
  (Ex. 1,2,3,4...) or (Ex. 10, 9, 8, 7...)
  Geometric sequences are when you multiply or divide. (Ex. 1, 10, 100, 1000...) or (Ex. 1000, 100, 10, 1)
  I hope this helps :)
  Comment on MSTC_Grinding's post “Yep. Arithmetic sequences...”
  (2 votes)
ADITYA ROY
Posted 6 months ago. Direct link to ADITYA ROY's post “Can anyone here explain i...”
Can anyone here explain in detail , what is the meaning of Recursive ?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted 6 months ago. Direct link to Kim Seidel's post “Use any dictionary websit...”
  Use any dictionary website to get the formal definition.
  
  With the recursive equation for a sequence, you must know the value of the prior term to create the next term. So, you follow a repetitive sequence of steps to get to the value you want. For example, to find the 4th term of a sequence using a recursive equation, you:
  1) Calculate the 1st term (this is often given to you).
  2) Use the value of the 1st term to calculate the 2nd term.
  3) Use the value of the 2nd term to calculate the 3rd term.
  4) Use the value of the 3rd term to calculate the 4th term.
  
  Basically, you can't get to the 4th term in one step. You have to go term to term up to the 4th term (or what ever term you want). If you want to get to the term directly, the explicit equation for a sequence would do that.
  Button navigates to signup page
  (4 votes)