Sequences That was easy A sequence is an ordered list of numbers that often form a pattern. Each number in the list is called a term of the sequence. A wooden post-and-rail fence is made as shown below. How many pieces of wood are needed to build a 4-section fence, a five-section fence, a six-section fence? I think I see a pattern. 1 section 4 pieces 2 sections 7 pieces 3 sections 10 pieces 6-section fence + 3 + 3 4-section fence 19 pieces 5-section fence 13 pieces That was easy 16 pieces

Extending Sequences That was easy Describe a pattern in each sequence. Determine what the next two terms are. 5, 8, 11, 14,… 17, 20 2, -4, 8, -16,… 32, -64 + 3 + 3 + 3 * -2 * -2 * -2 3, 6, 12, 24,… 48, 96 8, 3, -2, -7,… -12, -17 * 2 * 2 * 2 - 5 - 5 - 5 400, 200, 100, 50,… 25, 12.5 * ½ * ½ * ½ That was easy

Arithmetic Sequences Asi De Facil In an arithmetic sequence the difference between consecutive terms is constant. An arithmetic sequence follows a pattern of adding a fixed amount from one term to the next. This difference is called the common difference. 3, 8, 13, 18,… 9, 2, -5, -12,… 5, 9, 12, 14,… + 5 + 5 + 5 - 7 - 7 - 7 + 4 + 3 + 2 There is a common difference of 5, so it is arithmetic. There is a common difference of -7, so it is arithmetic. There is no common difference, so it is not arithmetic. 2, 6, 18, 54,… That’s not a common difference. That’s not adding. *3 *3 *3 There is no common difference derived by addition, so it is not arithmetic. Asi De Facil

Identifying Arithmetic Sequences Tell whether the sequence is arithmetic. If it is, what is the common difference? 8, 15, 22, 30,… 0.2, 1.5, 2.8, 4.1,… 7, 9, 11, 13,… + 7 + 7 + 8 + 1.3 + 1.3 + 2 + 2 + 2 + 1.3 No Yes CD = + 1.3 Yes CD = +2 10, 4, -2, -8,… 2, -2, 2, -2,… 2, 11, 21, 32,… -6 -6 -6 * -1 * -1 * -1 + 9 + 10 + 11 Yes CD = -6 No No

Practice with Sequences a) Describe a pattern in each sequence. b) Determine what the next two terms are. c) Explain why the sequence is or is not arithmetic. 7, 3, -1, -5,… 2, 4, 8, 16,… - 4 - 4 - 4 * 2 * 2 * 2 a) Add negative 4 c) It is arithmetic because the common difference is 4 and the pattern is addition. a) Multiply by 2 c) It is not arithmetic because there is not a common difference derived by addition. b) -9, -13 b) 32, 64 1, -3, 9, -27,… 12, 19, 26, 33,… * -3 * -3 * -3 +7 +7 +7 a) Multiply by negative 3 c) It is not arithmetic because there is not a common difference derived by addition. a) Add 7 c) It is arithmetic because the common difference is 7 and the pattern is addition. b) 40, 47 b) 81, -243

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Recursive Formula A recursive formula is a function rule that relates each term of a sequence after the first term to the ones before it. 3, 11, 19, 27,… Let n = the term number in the sequence Let A(n) = the value of the nth term in the sequence CD = +8 Value of term 1 = A(1) = 3 The recursive formula for this sequence is Value of term 2 = A(2) = A(1) + 8 = 3 + 8 = 11 Value of term 3 = A(3) = A(2) + 8 = 11 + 8 = 19 Value of term 4 = A(4) = A(3) + 8 = 19 + 8 = 27 Value of term n = A(n) = A(n - 1) + 8

Writing a Recursive Formula Write a recursive formula for the given sequence 23, 35, 47, 59,… 97, 88, 79, 70,… That was easy CD = +12 CD = -9 A(1) = 23 A(1) = 97 A(2) = A(1) + 12 = 23 + 12 = 35 A(2) = A(1) - 9 = 97 - 9 = 88 A(3) = A(2) + 12 = 35 + 12 = 47 A(3) = A(2) - 9 = 88 - 9 = 79 A(4) = A(3) + 12 = 47 + 12 = 59 A(4) = A(3) - 9 = 79 - 9 = 70 A(n) = A(n - 1) + 12 A(n) = A(n - 1) - 9

More Writing Recursive Formulas Write a recursive formula for the given sequence 70, 77, 84, 91,… 4.6, 4.7, 4.8, 4.9,… CD = +7 CD = +0.1 Asi De Facil 13, 10, 7, 4,… 13, 5, -3, -11,… CD = -3 CD = -8

Explicit Formula 16, 27, 38, 49,… CD = +11 An explicit formula is a function rule that relates each term of a sequence to the term number nth term first term term number common difference 16, 27, 38, 49,… CD = +11

Writing an Explicit Formula from a Recursive Formula

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Writing an Recursive Formula from a Explicit Formula first term term number common difference

More Writing an Recursive Formula from a Explicit Formula That was easy

Finding Specific Terms with an Explicit Formula Find the third, fifth, and thirteenth terms of the sequence described by each explicit formula.

More Finding Specific Terms with an Explicit Formula Find the third, fifth, and thirteenth terms of the sequence described by each explicit formula.

Homework Page 279 - 280: 42 - 52 Even Numbers