2012 Pearson Education, Inc. Slide 1-2-1 Chapter 1 The Art of Problem Solving.
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Transcript of 2012 Pearson Education, Inc. Slide 1-2-1 Chapter 1 The Art of Problem Solving.
![Page 1: 2012 Pearson Education, Inc. Slide 1-2-1 Chapter 1 The Art of Problem Solving.](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649f445503460f94c65038/html5/thumbnails/1.jpg)
2012 Pearson Education, Inc. Slide 1-2-1
Chapter 1Chapter 1The Art of The Art of Problem SolvingProblem Solving
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2012 Pearson Education, Inc. Slide 1-2-2
Chapter 1: Chapter 1: The Art of Problem SolvingThe Art of Problem Solving
1.1 Solving Problems by Inductive Reasoning
1.2 An Application of Inductive Reasoning: Number Patterns
1.3 Strategies for Problem Solving
1.4 Calculating, Estimating, and Reading Graphs
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2012 Pearson Education, Inc. Slide 1-2-3
Section 1-2Section 1-2
An Application of Inductive Reasoning: Number Patterns
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2012 Pearson Education, Inc. Slide 1-2-4
An Application of Inductive An Application of Inductive Reasoning: Number PatternsReasoning: Number Patterns
• Number Sequences
• Successive Differences
• Number Patterns and Sum Formulas
• Figurate Numbers
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2012 Pearson Education, Inc. Slide 1-2-5
Number SequencesNumber Sequences
Number SequenceA list of numbers having a first number, a second number, and so on, called the terms of the sequence.
Arithmetic SequenceA sequence that has a common difference between successive terms.
Geometric SequenceA sequence that has a common ratio between successive terms.
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2012 Pearson Education, Inc. Slide 1-2-6
Successive DifferencesSuccessive Differences
Process to determine the next term of a sequence using subtraction to find a common difference.
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2012 Pearson Education, Inc. Slide 1-2-7
14, 22, 32, 44,...
14 22 32 44
8 10 12 Find differences
2 2 Find differences
Example: Successive DifferencesExample: Successive Differences
Use the method of successive differences to find the next number in the sequence.
Build up to next term: 58
2
14
58
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2012 Pearson Education, Inc. Slide 1-2-8
Number Patterns and Sum FormulasNumber Patterns and Sum Formulas
Sum of the First n Odd Counting NumbersIf n is any counting number, then
Special Sum FormulasFor any counting number n,
( 1)and 1 2 3 .
2
n nn
2 3 3 3(1 2 3 ) 1 2 n n
21 3 5 (2 1) . n n
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2012 Pearson Education, Inc. Slide 1-2-9
Example: Sum FormulaExample: Sum Formula
Use a sum formula to find the sum
1 2 3 48.
Solution ( 1)
1 2 32
n nn
with n = 48:
48(48 1)1176.
2
Use the formula
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2012 Pearson Education, Inc. Slide 1-2-10
Figurate NumbersFigurate Numbers
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2012 Pearson Education, Inc. Slide 1-2-11
Formulas for Triangular, Square, and Formulas for Triangular, Square, and Pentagonal NumbersPentagonal Numbers
For any natural number n, ( 1)
the th triangular number is given by ,2n
n nn T
2the th square number is given by , andnn S n
(3 1)the th pentagonal number is given by .
2
nn n
n P
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2012 Pearson Education, Inc. Slide 1-2-12
Example: Figurate NumbersExample: Figurate Numbers
Use a formula to find the sixth pentagonal number
Solution (3 1)
2nn n
P
with n = 6:6
6[6(3) 1]51.
2P
Use the formula