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Chapter 1 The Art of Problem Solving 2012 Pearson Education, Inc. Slide 1-2-1 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. Slide 1-2-2 Section 1-2 An Application of Inductive Reasoning: Number Patterns 2012 Pearson Education, Inc. Slide 1-2-3 An Application of Inductive Reasoning: Number Patterns • • • • Number Sequences Successive Differences Number Patterns and Sum Formulas Figurate Numbers 2012 Pearson Education, Inc. Slide 1-2-4 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. Slide 1-2-5 Successive Differences Process to determine the next term of a sequence using subtraction to find a common difference. 2012 Pearson Education, Inc. Slide 1-2-6 Example: Successive Differences Use the method of successive differences to find the next number in the sequence. 14, 22, 32, 44,... 14 22 8 32 10 2 14 12 2 58 44 2 Find differences Find differences Build up to next term: 58 2012 Pearson Education, Inc. Slide 1-2-7 Number Patterns and Sum Formulas Sum of the First n Odd Counting Numbers If n is any counting number, then 2 1 3 5 (2n 1) n . Special Sum Formulas For any counting number n, (1 2 3 n)2 13 23 n3 n(n 1) and 1 2 3 n . 2 2012 Pearson Education, Inc. Slide 1-2-8 Example: Sum Formula Use a sum formula to find the sum 1 2 3 48. Solution Use the formula 1 2 3 with n = 48: n(n 1) n 2 48(48 1) 1176. 2 2012 Pearson Education, Inc. Slide 1-2-9 Figurate Numbers 2012 Pearson Education, Inc. Slide 1-2-10 Formulas for Triangular, Square, and Pentagonal Numbers For any natural number n, n(n 1) the nth triangular number is given by Tn , 2 the nth square number is given by Sn n2 , and n(3n 1) the nth pentagonal number is given by Pn . 2 2012 Pearson Education, Inc. Slide 1-2-11 Example: Figurate Numbers Use a formula to find the sixth pentagonal number Solution n(3n 1) Use the formula Pn 2 with n = 6: 6[6(3) 1] P6 51. 2 2012 Pearson Education, Inc. Slide 1-2-12