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The Art of Problem Solving: Inductive Reasoning and Number Patterns

This chapter explores the use of inductive reasoning to solve problems, with a focus on number patterns. It covers strategies for problem solving, calculating and estimating, and reading graphs. It also introduces arithmetic and geometric sequences, successive differences, number patterns, and sum formulas.

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The Art of Problem Solving: Inductive Reasoning and Number Patterns

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  1. Chapter 1 The Art of Problem Solving 2012 Pearson Education, Inc.

  2. Chapter 1: The 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 2012 Pearson Education, Inc.

  3. Section 1-2 • An Application of Inductive Reasoning: Number Patterns 2012 Pearson Education, Inc.

  4. An Application of Inductive Reasoning: Number Patterns • Number Sequences • Successive Differences • Number Patterns and Sum Formulas • Figurate Numbers 2012 Pearson Education, Inc.

  5. Number Sequences Number Sequence A list of numbers having a first number, a second number, and so on, called the terms of the sequence. Arithmetic Sequence A sequence that has a common difference between successive terms. Geometric Sequence A sequence that has a common ratio between successive terms. 2012 Pearson Education, Inc.

  6. Successive Differences Process to determine the next term of a sequence using subtraction to find a common difference. 2012 Pearson Education, Inc.

  7. Example: Successive Differences Use the method of successive differences to find the next number in the sequence. 14, 22, 32, 44,... 14 22 32 44 8 10 12 Find differences 2 2 Find differences 58 14 2 Build up to next term: 58 2012 Pearson Education, Inc.

  8. Number Patterns and Sum Formulas Sum of the First n Odd Counting Numbers If n is any counting number, then Special Sum Formulas For any counting number n, 2012 Pearson Education, Inc.

  9. Example: Sum Formula Use a sum formula to find the sum Solution Use the formula with n = 48: 2012 Pearson Education, Inc.

  10. Figurate Numbers 2012 Pearson Education, Inc.

  11. Formulas for Triangular, Square, and Pentagonal Numbers For any natural number n, 2012 Pearson Education, Inc.

  12. Example: Figurate Numbers Use a formula to find the sixth pentagonal number Solution Use the formula with n = 6: 2012 Pearson Education, Inc.

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