Babinec Senior Research Presentation

Patterns in Pascal’s Triangle:Do They Apply to Similar Triangular Arrays?: 

Patterns in Pascal’s Triangle: Do They Apply to Similar Triangular Arrays? Nicole Forcum Mentor: Dr. Scott Sportsman Faculty: Dr. Lisa Rome Senior Research April 2005

Topics to Discuss:: 

Topics to Discuss: History Defining Pascal’s Triangle Properties Proof techniques Where to go now

History of Pascal’s Triangle: 

History of Pascal’s Triangle Who is given credit? A Treatise on the Arithmetic Triangle A Treatise on the Arithmetical Triangle Uses of the Arithmetical Triangle Figurate numbers Theory of combinations Dividing the stake in games of chance Finding powers of binomial expressions

Pascal’s Triangle: 

Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1

Defining Pascal’s Triangle: 

Defining Pascal’s Triangle an,r = position value n represents row number r represents element Definition: an,r = an-1,r-1 + an-1,r an,n = 1, an,0 = 1 1 1 1 1 2 1 1 3 3 1

Properties of Pascal’s Triangle: 

Properties of Pascal’s Triangle Hockey Stick Sum of Rows Alternating Sums Hexagon Pattern

Properties of Pascal’s Triangle:Hockey Stick: 

Properties of Pascal’s Triangle: Hockey Stick 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1

Multiple Triangular Array:Hockey Stick: 

Multiple Triangular Array: Hockey Stick 2 2 2 2 4 2 2 6 6 2 2 8 12 8 2 2 10 20 20 10 2 2 12 30 40 30 12 2 2 14 42 70 70 42 12 2 2 16 56 112 140 112 56 16 2

Consecutive Triangular Array:Hockey Stick Pattern: 

Consecutive Triangular Array: Hockey Stick Pattern 1 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7 1 8 28 56 70 56 28 8

Consecutive Triangular Array:Hockey Stick Pattern: 

Consecutive Triangular Array: Hockey Stick Pattern 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1

Powers of 2 Triangular Array:Hockey Stick: 

Powers of 2 Triangular Array: Hockey Stick 1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 64 1 8 29 64 99 120 127 128 1 9 37 93 163 219 247 255 256

Properties of Pascal’s Triangle: Sum of Rows: 

Properties of Pascal’s Triangle: Sum of Rows 1 = 20 1+1 = 2 = 21 1+2+1 = 4 = 22 1+3+3+1 = 8 = 23 1+4+6+4+1 =16 = 24 1+5+10+10+5+1 = 32 =25 1+6+15+20+15+6+1 = 64 = 26 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Consecutive Triangular Array:Sum of Rows: 

Consecutive Triangular Array: Sum of Rows 1+2 = 3 = 22 - 1 1+3+3 = 7= 23 - 1 1+4+6+4 = 15 = 24 - 1 1+5+10+10+5 = 31 = 25 - 1 1+6+15+20+15+6 = 63 = 26 - 1 1+7+21+35+35+21+7 = 127 = 27 - 1 1 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7

Powers of 2 Triangular Array:Sum of Rows: 

Powers of 2 Triangular Array: Sum of Rows 1+2 = 3 1+3+4 = 8 1+4+7+8 = 20 1+5+11+15+16 = 48 1+6+16+26+31+32 = 112 1+7+22+42+57+63+64 = 256 1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 64

Why?: 

Why? n = 4 : 1 5 11 15 16 1 + 5 + 11 + 15 + 16 = s 16 + 15 + 11 + 5 + 1 = s 16 + 16 + 16 + 16 + 16 + 16 = 2s 6*24 = 2s 6*23 = s (n+2)2n-1 = s

Properties of Pascal’s Triangle:Alternating Row Sums: 

Properties of Pascal’s Triangle: Alternating Row Sums 1-1 = 0 1-3+3-1 = 0 1-5+10-10+5-1 = 0 1-2+1 = 0 1-4+6-4+1 = 0 1-6+15-20+15-6+1 = 0 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Consecutive Triangular Array:Alternating Row Sums: 

Consecutive Triangular Array: Alternating Row Sums 1-2 = -1 1-3+3 = 1 1-4+6-4 = -1 1-5+10-10+5 = 1 1-6+15-20+15-6 = -1 1 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6

Powers of 2 Triangular Array:Alternating Row Sums: 

Powers of 2 Triangular Array: Alternating Row Sums 1-2 = -1 = -(20) 1-3+4 = 2 = 21 1-4+7-8 = -4 = -(22) 1-5+11-15+16 = 8 = 23 1-6+16-26+31-32 = -16 = -(24) 1-7+22-42+57-63+64 = 32 = 25 1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 64

Properties of Pascal’s Triangle:Hexagon Pattern: 

Properties of Pascal’s Triangle: Hexagon Pattern 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1

Consecutive Triangular Array:Hexagon Pattern: 

Consecutive Triangular Array: Hexagon Pattern 1 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7 1 8 28 56 70 56 28 8

Powers of 2 Triangular Array:Hexagon Pattern: 

Powers of 2 Triangular Array: Hexagon Pattern 1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 64 1 8 29 64 99 120 127 128 1 9 37 93 163 219 247 255 256

Counting Proof: an,r = # ways to choose r from nFactorial Proof: an,r =Decision Proof : an,r = # of paths, L & R Movements: 

Counting Proof: an,r = # ways to choose r from n Factorial Proof: an,r = Decision Proof : an,r = # of paths, L andamp; R Movements

Decision Proof:Left & Right Movements: 

Decision Proof: Left andamp; Right Movements a4,0 = 1 a4,4 = 1 a3,2 = 3 a4,2 = 6

How Will I Apply The Research?: 

How Will I Apply The Research? High School Teacher Lesson plans

Thank You : 

Thank You Parents andamp; Dan Classmates Dr. Sportsman andamp; Dr. Rome

Questions?: 

Questions?