Arithmetic Sequences

Indicators: PFA#1,2

Created by Anny Lin, Crestwood Middle School

•Recognize arithmetic sequences•Extend and write formulas for arithmetic sequences

A sequence is a set of numbers called terms, in a specific order.

If the difference between successive terms is constant, then it is called arithmetic sequence.

The difference between the terms is called the common difference.

Common Difference

Term:Arithmetic Sequence:

You can use the common difference of an arithmetic sequence to find

the next term in the sequence.http://www.basic-mathematics.com/images/sequence1.gif

In other words, an arithmetic sequence is a numerical pattern that increases or decreases at a constant rate or value called the common difference.

3, 7, 11, 15,…

2, 6, 10, 14,…13, 9, 5, 1,…16, 12, 9, 8, 4, 2,…What about -17, -8, 3,14,25,…?

This is Arithmetic.

Arithmetic

Not Arithmetic

Is the following sequence arithmetic?

1. Yes

2. No

0.7, 0.5, 0.3, 0.1,…

The common difference is -0.2.

1. 3, 6, 12, 24,…

2. -7, -3, 1, 5,...

3. 1/5, 1/7, 1/9, 1/11,...

4. -10, 5, -5/2, 5/4,...

Which sequence is an arithmetic sequence?

1. -7, 0, 7, 14,…

2. 0, 1/2, 1, 3/2,...

3. 10, 6, 2, -2,...

4. 2, 4, 8, 16,...

Which sequence is NOT an arithmetic sequence?

Is the following sequence arithmetic?

1. Yes

2. No

1/2, 1/4, 1/8, 1/16,…

Is the following sequence arithmetic?

1. Yes

2. No

1 2 4 5, , 1, , , ...

3 3 3 3

You can use the common difference of an arithmetic sequence to find

the next term in the sequence.

Find the next three terms of the arithmetic sequence.

1) -15, -13, -11, -9,…

What is the common difference?

2

What is the next term?

-7

What are the next three terms?

-7, -5, -3

Find the next three terms of the arithmetic sequence.

1) 1/8, 1/4, 3/8 ,1/2 ,…

What is the common difference?

1/8

What is the next term?

5/8

What are the next three terms?

5/8, ¾, 7/8

Real-World Example

BOOKS The arithmetic sequence 36, 39, 42, 45, … represents the number of books in Jacob’s collection at the end of each month. Find the next three terms.Find the common difference by subtracting successive terms.

The common difference is +3.Add 3 to the last term of the sequence to get the next term in the sequence. Continue adding 3 until the next three terms are found.

• P 168 #1-4, 12-19 all

• Worksheet for third period

Words: Each term of an arithmetic sequence after the first term can be found by adding the common difference to the preceding term.

Writing Arithmetic Sequence

Symbols: An arithmetic sequence, a1 , a2 ,…, can be found as following:a1, a2 =a1 +d, a3 =a2 +d, a4 = a3 +d…,Where d is the common differences a1 is the first term, a2 is the second term, and so on.

Given the arithmetic sequence 3, 6, 9, 12, 15,…

Find a1, a2, a3, a4.

a1 =3

a2 =6

a3 =9

a4 =12

What is a5 ?

a5 =15

What about a6 ?

a6 =18

What is the common difference?

3

Given the arithmetic sequence 14, 28, 42, 56,…

Find a1, a2, a3, a4.

a1 =14

a2 =28

a3 =42

a4 =56

What is a5 ?

a5 =70

What about a6 ?

a6 =84

What is the common difference?

14

a1=6

a4 = a3 +d

a2 =a1 +d=6+9=15

=15+9=24

a3 =a2 +d=24+9=33

Given a1=6 and common differences: d=9. What is a2?

a1=24 common differences: d=-8

For example:

a4 = a3 +d

a2 =a1 +d=24-8=16

=16-8=8

a3 =a2 +d

=8-8=0

25, 23, 21, 19, …,

What is a1 ?

What is a2 ?

25

23

What is the common difference d?

d=-2

a2 =a1 +d

=25+(-2)

=23

25, 23, 21, 19, …,

What is a3 ?

=21

d=-2

a3 =a2 +d = a1 +d+d a3= a1 +2d =25+2(-2) =25-4

What is a4 ?

a4 = a3+d = a1 +d+d+d a4 = a1 +3d =25+3(-2) =25-6 =19

Watch Out!!!

• Do you see the pattern??

What is the relationship between a4 and 3d?

What is the relationship between a3 and 2d?

What is the relationship between an and (n-1)d?

25, 27, 29, 31, …, d=2

an =a1 +( )d = a1 +(n-1)d =25 +(n-1)(2)

What is an ?

What is a4 ?

a4 = a3+ d = a2+ d+d = a1 + d+d+d a4 = a1 +3d =25+3(2) =25 + 6 =31

Use the smart board

Term Symbol In terms of a1 and d

numbers

First term a1 a1 5

Second term a2 a1 +d 5+1(2)=7

Third term a3 a2 +d=(a1 +d)+d=a1 +2d

5+2(2)=9

Fourth term a4 a3 +d=(a1 +2d)+d=a1 +3d

5+3(2)=11

: : : :

nth term an an +d=a1 +(n-1)d

5+(n-1)(2)

Formula

The nth term an of an arithmetic sequence with first term a1 and common difference d is given by

an =a1 +(n-1)d.

Where n is a positive integer.