Sequences in GeoGebra

MSP SI 2007 Sequences Joan Carter

Sequences in GeoGebra

Sequences


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.


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


Number Patterns

Find the next two terms of each sequence. Describe how you found each term.

0, 1, 3, 6, 10, 15, ___, ___

11, 22, 33, 44, 55, ___, ___

5, 8, 7, 10, 9, 12, 11, __,__

6666 7777

2121 2828

1414 1313

Slide Courtesy of Guy Barmoha


Sequences

€

(a,ar1,ar

2,ar

3...)

€

F(n) :=

0

1

F(n −1) + F(n − 2)

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

Examples Sequence

Notation

€

(a1,a2,a3,...an )

€

(1,2,3,...)

€

(1,4,7,...)

€

arithmetic

€

(2,4,8,...)

€

geometric

€

(1,1,2,3,5,8...)

€

Fibonacci

€

(1,1

2,1

4,1

8,...)


Arithmetic Sequences

Sequence of numbers where any 2 successive members have a common difference

Example:( 0, 1, 2, 3, 4 )

+ 1 +1 +1 +1


Arithmetic Sequences

Sequence of numbers where any 2 successive members have a common difference

Example:( 0, 3, 6, 9, 12 )

+ 3 +3 +3 +3


What would these sequences look like if we graphed them?

X Y

0 0

1 3

2 6

3 9

4 12



X Y

0 1

1 4

2 7

3 10

4 13

A line?Possibly, but we need to checkit out! GeoGebra will help us.



X Y

0 1

1 4

2 7

3 10

4 13

seq_line1.ggb


Sequences

X Y

0 1

1 4

2 7

3 10

4 13

Yes, this is a linear sequence!How would we find the equationof the line without graphing?

Common difference = 1

Common difference = 3

Slope= change y = 3

change x 1 y = 3 x + ? y = 3 x + 1

y = m x + b


Number Sequences

Term

Value

1 2 3 4 5 6

4 7 10 13 16 19

7

What is the 7th term of this sequence?


22

… 200

… ?

Slide Courtesy of Guy Barmoha


Number Sequences

Term

Value

1 2 3 4 5 6

4 7 10 13 16 19

7



22

… 200

… ?

22

seq_line2.ggb


To find the nth term algebraically, usean = a1 + (n-1) d

a1 = initial term, d = common difference

.

Sequences

What equation is this? Slope-Intercept Form y = 3x + 1

y = 3(200) + 1 y = 601

Term

Value

1 2 3 4 5 6

4

7 … 200

7 19161310 …22 ?


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]


Sequences: Segments in GeoGebraSlide 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.

seq_line_art1.ggb


Segment Sequences

Markus’line art tool seq_line_art2.ggb


Sequences of Segments on a Circleseq_circle_segments1.ggb

seq_circle_segments3.ggb


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.

Documents

Sequences in GeoGebra