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Section 1.2 Patterns and Problem Solving Arithmetic Sequence An arithmetic sequence has a common difference between each of the numbers in the sequence. Examples: 1, 3, 5, 7, 9, … 1, 2, 3, 4, 5, … 2, 4, 6, 8, 10, … 2 Geometric Sequence A geometric sequence is formed by multiplying the previous number by a selected number. Therefore, there is a common ratio from one number to the next in a geometric sequence. Examples: 2, 4, 8, 16, 32, 64, … 3, 9, 27, 81, 243, … 3 22)Identify each of the sequences as arithmetic or geometric. State a rule for obtaining each number from the preceding number. What is the 12th number in each sequence? a) 280, 257, 234, 211,... b) 17, 51, 153,459,… 4 Method of Finite Differences Finding number patterns by looking at the differences between consecutive terms. 5 26)Use the method of finite differences to find the next number in each sequence. a) 3, 7, 13, 21, 31, 43,… b) 215, 124, 63, 26, 7, … 6 Fibonacci Numbers After the first two numbers, 1 and 1, each successive number is obtained by adding the previous two. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … 7 14) A Fibonacci-type sequence can be started with any two numbers. Find the missing numbers among the first 10 numbers of the Fibonacci-type sequences. a) 10, ___, 24, ___, ___, 100, ___, ___, ___, 686 b) 2, ___, ___, 16, 25, ___, ___, ___, ___, 280 8 14) continued c) The sum of the first 10 numbers in the sequence in part a) is equal to 11 times the seventh number, 162. What is this sum? d) Can the sum of the first 10 numbers in the sequence in part b) be obtained by multiplying the seventh number by 11? e) Do you think the sum of the first 10 numbers in any Fibonacci-type sequence will always be 11 times the seventh number? Try some other Fibonacci-type 9 sequences to support your conclusion. The previous problem is an example of inductive reasoning. Inductive reasoning is the process of forming conclusions on the basis of patterns, observations, examples, experiments. Otherwise known as an informed guess. 10 Pascal’s Triangle Row 0 1 2 3 4 5 6 . . 1 1 1 1 1 2 3 1 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 . . 11 20)Compute the sums of the numbers in the first few rows of Pascal’s triangle. What kind of sequence (arithmetic or geometric) do these sums form? 12 A counterexample is an example that shows a statement to be false. 40) Find a counterexample for each of the statements. a) Every whole number greater than 4 and less than 20 is the sum of two or more consecutive whole numbers. b) Every whole number between 25 and 50 is the product of two whole numbers greater than 1. 13