The Art of Problem Solving

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Presentation transcript:

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

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 Numeracy in Today’s World

An Application of Inductive Reasoning: Number Patterns Section 1-2 An Application of Inductive Reasoning: Number Patterns

An Application of Inductive Reasoning: Number Patterns Be able to recognize arithmetic and geometric sequences. Be able to apply the method of successive differences to predict the next term in a sequence. Be able to recognize number patterns. Be able to use sum formulas. Be able to recognize triangular, square, and pentagonal numbers.

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.

Example: Identifying Arithmetic or Geometric Sequences For each sequence, determine if it is an arithmetic sequence, a geometric sequence, or neither. If it is either arithmetic or geometric, give the next term in the sequence. a. 5, 10, 15, 20, 25, … b. 3, 12, 48, 192, 768, … Solution a. Subtract the terms and find that the common difference is 5. 10 – 5 = 5, 20 – 15 = 5. Arithmetic sequence; next term: 25 + 5 = 30

Example: Identifying Arithmetic or Geometric Sequences b. 3, 12, 48, 192, 768, … Solution a. If you divide any two successive terms, you find a common ratio of 4. Geometric sequence Next term: 768(4) = 3072

You Try!!

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

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

You Try!!

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,

Example

Another Example

You Try!!

Figurate Numbers

Formulas for Triangular, Square, and Pentagonal Numbers For any natural number n,

Example: Using the Formulas for Figurate Numbers Use a formula to find the sixth pentagonal number. Solution Use the formula: with n = 6: 18

You Try!!

Arithmetic Sequence Formula

You Try!!

Geometric Sequence Formula

You Try!!