Joan Carter MSP SI 2007Sequences Sequences in GeoGebra Sequences.

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.

Joan Carter MSP SI 2007Sequences Sequences in GeoGebra Sequences.

Documents

Transcript of Joan Carter MSP SI 2007Sequences Sequences in GeoGebra Sequences.