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Chapter 1Chapter 1The Art of The Art of Problem SolvingProblem Solving

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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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Section 1-2Section 1-2

An Application of Inductive Reasoning: Number Patterns

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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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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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Successive DifferencesSuccessive Differences

Process to determine the next term of a sequence using subtraction to find a common difference.

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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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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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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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Figurate NumbersFigurate Numbers

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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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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