Sequences in GeoGebra Sequences MSP SI 2007 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. MSP SI 2007 Sequences

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

21 28 Number Patterns 11, 22, 33, 44, 55, ___, ___ 66 77 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 Slide Courtesy of Guy Barmoha

Sequences Examples Sequence Notation MSP SI 2007 Sequences

Arithmetic Sequences Sequence of numbers where any 2 successive members have a common difference Example: ( 0, 1, 2, 3, 4 ) + 1 +1 +1 +1 MSP SI 2007 Sequences

Arithmetic Sequences Sequence of numbers where any 2 successive members have a common difference Example: ( 0, 3, 6, 9, 12 ) + 3 +3 +3 +3 MSP SI 2007 Sequences

What would these sequences look like if we graphed them? X Y 1 3 2 6 9 4 12 MSP SI 2007 Sequences

What would these sequences look like if we graphed them? X Y 1 4 2 7 3 10 13 A line? Possibly, but we need to check it out! GeoGebra will help us. MSP SI 2007 Sequences

What would these sequences look like if we graphed them? X Y 1 4 2 7 3 10 13 seq_line1.ggb MSP SI 2007 Sequences

Sequences X Y 1 4 2 7 3 10 13 Common difference = 1 Common Yes, this is a linear sequence! How would we find the equation of the line without graphing? X Y 1 4 2 7 3 10 13 y = m x + b Common difference = 1 Common difference = 3 Slope= change y = 3 change x 1 y = 3 x + ? y = 3 x + 1 MSP SI 2007 Sequences

Number Sequences What is the 7th term of this sequence? Value 1 2 3 4 5 6 7 … 200 ? 7 10 13 16 19 22 What is the 7th term of this sequence? What is the 200th term of this sequence? MSP SI 2007 Sequences Slide Courtesy of Guy Barmoha

Number Sequences 22 What is the 7th term of this sequence? Value 1 2 3 4 5 6 7 … 200 ? 7 10 13 16 19 22 22 What is the 7th term of this sequence? What is the 200th term of this sequence? seq_line2.ggb MSP SI 2007 Sequences

Sequences What equation is this? Slope-Intercept Form . To find the nth term algebraically, use an = a1 + (n-1) d a1 = initial term, d = common difference . Term Value 1 2 3 4 5 6 7 … 200 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

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

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

Segment Sequences Markus’ line art tool seq_line_art2.ggb MSP SI 2007

Sequences of Segments on a Circle seq_circle_segments1.ggb seq_circle_segments3.ggb MSP SI 2007 Sequences

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 joan.carter@browardschools.com Special thanks to Dr. Markus Hohenwarter and Guy Barmoha, MST. MSP SI 2007 Sequences