Math 2205 Section 1'1

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Math 2205 Section 1.1

• Explorations with patterns
• Sequences
• Arithmetic, geometric, other

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Example 1

• What is the next number?
• 2, 4, 6,

8
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Example 2

• What is the next number?
• 5, 10, 15,

20
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Example 3

• What is the next number?
• 4, 10, 16,

22
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What's the difference? Example 1

• 2, 4, 6,
• The difference between any two terms is ?
• 6 4 2
• 4 2 2
• So add 2 to get the next term.

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What's the difference? Example 2

• 5, 10, 15,
• The difference between any two terms is ?
• 15 10 5
• 10 5 5
• So add 5 to get the next term.

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What's the difference? Example 3

• 4, 10, 16,
• The difference between any two terms is ?
• 16 10 6
• 10 4 6
• So add 6 to get the next term.

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Example 4

• What is the next number?
• 2, 4, 8,

16
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Example 5

• What is the next number?
• 5, -25, 125,

-625
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What's the ratio? Example 4

• 2, 4, 8,
• The ratio between any two terms is ?
• 8 4 2
• 4 2 2
• So multiply by 2 to get the next term.

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What's the ratio? Example 5

• 5, -25, 125,
• The ratio between any two terms is ?
• 125 (-25) -5
• (-25) 5 -5
• So multiply by -5 to get the next term.

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Arithmetic and Geometric Sequences

• 2, 4, 6,
• 5, 10, 15,
• 4, 10, 16,
• 2, 4, 8,
• 5, -25, 125,

Arithmetic
Geometric
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Inductive Reasoning

• Inductive Reasoning Process of making a
  generalization based upon a limited number of
  observations or examples.
• Basis of experimental science (hypothesis) and a
  fundamental part of mathematics (patterns).
• Specific to general.

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Deductive Reasoning

• Deductive Reasoning Process of reaching a
  necessary conclusion from given facts or
  assumptions.
• Involves thinking logically (solve
  crimesSherlock Holmes).

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Mathematics and Inductive/Deductive Reasoning

• Mathematics uses inductive and deductive
  reasoning.
• First inductive reasoning is used to make a
  generalization about an observed pattern in a
  variety of examples.
• Then, using logical reasoning (deductive
  reasoning) demonstrates that the generalization
  must be true if certain assumptions are made.

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How many Ancestors?

• Parents
• Grand Parents
• Great Grand Parents
• G, G, Grand Parents
• G, G, G, Grand
• G, G, G, G, Grand

• 2
• 4
• 6
• 8
• 16
• 32
• 64
• 128

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31
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Mathematics and Inductive/Deductive Reasoning

• One counter example is enough to disprove a
  conjecture.
• What is going on with the circle, chords, and
  regions is more complicated than 2n.
• The On-Line Encyclopedia of Integer Sequences
• 1, 2, 4, 8, 16, 31

http//www.research.att.com/njas/sequences/
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1
2
3
3
6
4
10
5
15
6
21
Triangular Numbers
7
28
8
36
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Triangular Numbers
Recursive Formula T(n) T(n-1) n

• 1 1
• 1 2 1 2 3
• 1 2 3 3 3 6
• 1 2 3 4 6 4 10
• 1 2 3 4 5 10 5 15
• 1 2 3 4 5 6 15 6 21
• 1 2 3 4 5 6 7 21 7 28

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Add up the Consecutive Odd Numbers

• 0 1 1
• 1 3 4
• 4 5 9
• 9 7 16
• 16 9 25
• 25 11 36

What is the recursive formula?
Square Numbers
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Objectives