Arithmetic Sequences
•Recognize arithmetic
sequences
•Extend and write formulas
for arithmetic sequences
Indicators: PFA#1,2
Created by Anny Lin, Crestwood Middle School
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.
Term:
Arithmetic Sequence:
Common
Difference
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,…
Arithmetic
2, 6, 10, 14,…
13, 9, 5, 1,…
Not Arithmetic
16, 12, 9, 8, 4, 2,…
What about -17, -8, 3,14,25,…?
This is Arithmetic.
Is the following sequence
arithmetic?
0.7, 0.5, 0.3, 0.1,…
1. Yes
2. No
The common difference is -0.2.
1.
2.
3.
4.
Which sequence is an arithmetic
sequence?
3, 6, 12, 24,…
-7, -3, 1, 5,...
1/5, 1/7, 1/9, 1/11,...
-10, 5, -5/2, 5/4,...
1.
2.
3.
4.
Which sequence is NOT an
arithmetic sequence?
-7, 0, 7, 14,…
0, 1/2, 1, 3/2,...
10, 6, 2, -2,...
2, 4, 8, 16,...
Is the following sequence
arithmetic?
1/2, 1/4, 1/8, 1/16,…
1. Yes
2. No
Is the following sequence
arithmetic?
1 2
4 5
, , 1, , , ...
3 3
3 3
1. Yes
2. No
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
Writing Arithmetic Sequence
Words: Each term of an arithmetic
sequence after the first term can be
found by adding the common
difference to the preceding term.
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
Given a1=6 and common differences: d=9. What is a2?
a1=6
a2 =a1 +d
=6+9
=15
a3 =a2 +d
=15+9
=24
a4 = a3 +d
=24+9
=33
For example:
a1=24
a2 =a1 +d
=24-8
=16
a3 =a2 +d
=16-8
=8
common differences:
d=-8
a4 = a3 +d
=8-8
=0
25, 23, 21, 19, …,
What is a1 ?
25
What is a2 ?
23
a2 =a1 +d
=25+(-2)
=23
What is the common
difference d?
d=-2
25, 23, 21, 19, …,
What is a3 ?
a3 =a2 +d
= a1 +d+d
a3= a1 +2d
=25+2(-2)
=25-4
=21
d=-2
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 a3 and 2d?
What is the relationship
between a4 and 3d?
What is the relationship
between an and (n-1)d?
25, 27, 29, 31, …,
d=2
What is a4 ?
a 4 = a 3+ d
= a2+ d+d
= a1 + d+d+d
a4 = a1 +3d
=25+3(2)
=25 + 6
=31
What is an ?
an =a1 +( )d
= a1 +(n-1)d
=25 +(n-1)(2)
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.