Patterns in Pascals Triangle: Do They Apply to Similar Triangular Arrays

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Patterns in Pascals TriangleDo They Apply to
Similar Triangular Arrays?

• Nicole Forcum
• Mentor Dr. Scott Sportsman
• Faculty Dr. Lisa Rome
• Senior Research April 2005

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Topics to Discuss

• History
• Defining Pascals Triangle
• Properties
• Proof techniques
• Where to go now

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History of Pascals 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

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Pascals 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
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Defining Pascals 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
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Properties of Pascals Triangle

• Hockey Stick
• Sum of Rows
• Alternating Sums
• Hexagon Pattern

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Properties of Pascals TriangleHockey 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
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Multiple Triangular ArrayHockey Stick
2 2 2 2 4 2 2 6 6 2 2 8 12 8 2 2 10 20 20 10 2 2 1
2 30 40 30 12 2 2 14 42 70 70 42 12 2 2 16 56 112
140 112 56 16 2
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Consecutive Triangular ArrayHockey 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
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Consecutive Triangular ArrayHockey 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 1
5 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28
8 1
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Powers of 2 Triangular ArrayHockey 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
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Properties of Pascals Triangle Sum of Rows
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

• 14641 16 24
• 15101051 32 25
• 1615201561 64 26

• 1 20
• 11 2 21
• 121 4 22
• 1331 8 23

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Consecutive Triangular ArraySum of Rows
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

• 12 3 22 - 1
• 133 7 23 - 1
• 1464 15 24 - 1
• 1510105 31 25 - 1
• 161520156 63 26 - 1
• 17213535217 127 27 - 1

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Powers of 2 Triangular ArraySum of Rows
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

• 12 3
• 134 8
• 1478 20
• 15111516 48
• 1616263132 112
• 172242576364 256

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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
• 624 2s
• 623 s
• (n2)2n-1 s

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Properties of Pascals TriangleAlternating Row
Sums
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

• 1-1 0
• 1-33-1 0
• 1-510-105-1 0

• 1-21 0
• 1-46-41 0
• 1-615-2015-61 0

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Consecutive Triangular ArrayAlternating Row Sums
1 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6

• 1-2 -1
• 1-33 1
• 1-46-4 -1
• 1-510-105 1
• 1-615-2015-6 -1

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Powers of 2 Triangular ArrayAlternating Row Sums
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

• 1-2 -1 -(20)
• 1-34 2 21
• 1-47-8 -4 -(22)
• 1-511-1516 8 23
• 1-616-2631-32 -16 -(24)
• 1-722-4257-6364 32 25

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Properties of Pascals TriangleHexagon 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
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Consecutive Triangular ArrayHexagon 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
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Powers of 2 Triangular ArrayHexagon 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
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Counting Proof an,r ways to choose r from
nFactorial Proof an,r Decision Proof
an,r of paths, L R Movements

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Decision ProofLeft Right Movements
a4,0 1
a4,4 1
a3,2 3
a4,2 6
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How Will I Apply The Research?

• High School Teacher
• Lesson plans

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Thank You

• Parents Dan
• Classmates
• Dr. Sportsman Dr. Rome

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Questions?