Transcript Document

Sequences in GeoGebra
Sequences
MSP SI 2007
Sequences
Joan Carter
Sequences
What is a sequence?
An ordered list of objects (or events)
Like a set, it contains members
(called elements or terms) and the
number of terms is called the length.
MSP SI 2007
Sequences
Joan Carter
Workshop Objectives
You will be able to identify various
sequences and use GeoGebra to:
•
•
Graphically represent sequences
Use the sequence command to
create lists of objects
• Use the element command to find
the nth term of a sequence
•
Use the segment command to
create line designs
MSP SI 2007
Sequences
Joan Carter
Number Patterns
Find the next two terms of each sequence.
Describe how you found each term.
11, 22, 33, 44, 55, ___,
66 ___
77
21 ___
28
0, 1, 3, 6, 10, 15, ___,
14 13
5, 8, 7, 10, 9, 12, 11, __,__
MSP SI 2007
Sequences
Joan Carter
Slide Courtesy of Guy Barmoha
Sequences
Examples
(1,2,3,...)
(1,4,7,...)
Sequence

arithmetic (a1,a2,a3 ,...an )
(2,4,8,...)




MSP SI 2007
1 1 1
(1, , , ,...)
2 
4 8
Notation
geometric

(a,ar1,ar 2 ,ar 3 ...)
0


(1,1,2,3,5,8...) Fibonacci F(n) : 1

F(n 1)  F(n  2)




Sequences

Joan Carter
Arithmetic Sequences
Sequence of numbers where any 2
successive members have a common
difference
Example:
( 0, 1,
+ 1
MSP SI 2007
2,
+1
Sequences
3,
+1
4 )
+1
Joan Carter
Arithmetic Sequences
Sequence of numbers where any 2
successive members have a common
difference
Example:
( 0, 3,
+ 3
MSP SI 2007
6,
+3
Sequences
9,
+3
12 )
+3
Joan Carter
What would these sequences look
like if we graphed them?
X
0
1
2
3
4
MSP SI 2007
Y
0
3
6
9
12
Sequences
Joan Carter
What would these sequences look
like if we graphed them?
X
0
1
2
3
4
MSP SI 2007
Y
1
4
7
10
13
Sequences
A line?
Possibly, but we need to check
it out! GeoGebra will help us.
Joan Carter
What would these sequences look
like if we graphed them?
X
0
1
2
3
4
MSP SI 2007
Y
1
4
7
10
13
seq_line1.ggb
Sequences
Joan Carter
Sequences
Yes, this is a linear sequence!
How would we find the equation
of the line without graphing?
Common
difference = 1
MSP SI 2007
X
0
1
2
3
4
Y
1
4
7
10
13
Sequences
y = m x + b
Common
difference = 3
Slope= change y = 3
change x 1
y = 3 x + ?
y = 3 x + 1 Joan Carter
Number Sequences
Term
1
2
3
4
5
6
7
Value
4
7
10
13
16
19
22
… 200
…
?
What is the 7th term of this sequence?
What is the 200th term of this sequence?
MSP SI 2007
Sequences
Joan Carter
Slide Courtesy of Guy Barmoha
Number Sequences
Term
1
2
3
4
5
6
7
Value
4
7
10
13
16
19
22
… 200
…
?
What is the 7th term of this sequence?
22
What is the 200th term of this sequence?
seq_line2.ggb
MSP SI 2007
Sequences
Joan Carter
Sequences
To find the nth term algebraically, use
an = a1 + (n-1) d
a1 = initial term, d = common difference
Term
1
2
3
4
5
6
7
…
200
Value
4
7
10
13
16
19
22
…
?
.
What equation is this?
Slope-Intercept Form
y = 3x + 1
y = 3(200) + 1
y = 601
MSP SI 2007
Sequences
Joan Carter
Sequences: GeoGebra Review
To create a list of objects:
Use sequence command:
Sequence[expression e, variable i, number a, number b]
To find the nth element in a list:
Use element command:
Element[List L, number n]
MSP SI 2007
Sequences
Joan Carter
Sequences: Segments in GeoGebra
Slide background resembles Bezier curve
Dr. Pierre Bezier (1910-1999)
Engineer for French automaker
“Best fit” curve for manufacturing
Used in computer graphics
He used 4 points; We’ll use 3.
MSP SI 2007
Sequences
seq_line_art1.ggb
Joan Carter
Segment Sequences
Markus’
line art tool
seq_line_art2.ggb
MSP SI 2007
Sequences
Joan Carter
Sequences of Segments on a Circle
seq_circle_segments1.ggb
seq_circle_segments3.ggb
MSP SI 2007
Sequences
Joan Carter
Sequences
• SSS: MA.D.1.3.1, MA.D.2.4.1
• All files will be posted on tiki at
http://nsfmsp.fau.edu/tiki/tiki-index.php
• Contact me at [email protected]
• Special thanks to Dr. Markus Hohenwarter
and Guy Barmoha, MST.
MSP SI 2007
Sequences
Joan Carter