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The Art of Problem Solving: Number Patterns and Strategies for Success

Learn how to solve problems through inductive reasoning, number patterns, and strategies. Explore number sequences, successive differences, and sum formulas for counting numbers. Master the art of problem solving in this comprehensive guide.

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The Art of Problem Solving: Number Patterns and Strategies for Success

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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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