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Lesson 10.1: Sequences & Summation Notation Sequences –Algebraic formulafirst n terms –Factorial notation Series –Summation or Sigma notation

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1. 5, 9, 13, 17, _____ 2. -1, 1, -1, 1, -1, 1, _____ 3. 1, 1, 2, 3, 5, 8, 13, 21, ______ 4. 10000, - 1000, 100, - 10, 1, ______ 5. a + b, 2b, - a + 3b, - 2a + 4b, _________ Sequence: an ordered collection of comma-separated terms

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An infinite sequence { a n } is a function whose domain is the set of positive integers*. The function values a 1, a 2, a 3, a 4,... a n,... are the terms of the sequence. If the domain of the sequence consists of only the first n positive integers, then the sequence is a finite sequence. [p 656] 5, 9, 13, 17,..., a n,... Example: infinite sequence a1a1 first term What is a 6 ? ________ anan n th term Occasionally, sequences start with n=0

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Find the first four terms of the indicated sequence. 1. a n = 3n – 2 a 1 = 3(1) – 2 = a 2 = 3(2) – 2 = a 3 = 3(3) – 2 = a 4 = sequence: Algebraic formulafirst n terms

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Find the first four terms of the indicated sequence.

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Practice: p 664 # 7 & 10 Write the first five terms of the indicated sequence.

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Write an expression for the apparent n th term of the sequence 1. 1, 3, 5, 7,... n: 1 2 3 4... n terms: 1 3 5 7... a n Answer: a n = 2n - 1 Algebraic formulafirst n terms

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Write an expression for the apparent n th term of the sequence 2. 2, 5, 10, 17,... n: 1 2 3 4... n terms:... a n Answer:

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n: 1 2 3 4 5 n terms: a n Answer: a n =

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Factorial Notation [p 658] If n is a positive integer, then n factorial is defined by n! = 1 · 2 · 3 · 4 · 5 · · · (n – 1) · n Special case: 0! = 1 10! = 1 · 2 · 3 · 4 · 5 · · ·10 =3,628,800 4! = 1 · 2 · 3 · 4 = 24 On graphing calculators: On Ti89: MATH Probability ! On Ti84: MATH PRB !

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Find the first five terms of the sequence whose n th term is. Begin with n = 0. sequence:

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Fractions involving factorials can often be simplified. 1. 2. 3. 4.

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Series: the summation of the terms of a sequence The sum of the first n terms of a sequence is represented by where i is the index of summation, n is the upper limit of summation and 1 is the lower limit of summation*. [p 660] This is often referred to as sigma notation. The index does not have to i and the lower index of summation does not have to be 1.

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Properties of Sums 4(1) + 4(2) + 4(3) + 4(4) + 4(5) = 60 (2 2 + 2) + (3 2 + 2) + (4 2 + 2) + (5 2 + 2) + (6 2 + 2) = 100 c is any constant

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Use Sigma Notation to write a sum Similar to writing on expression for the apparent n th term. Place in sigma notation, specifying summation boundaries. n: 1 2 3 4 5 n terms: 1 4 7 10 13 a n a n = 3n - 2 1. 1 + 4 + 7 + 10 + 15 2. 3 – 9 + 27 – 81 + 243 - 729 n: 1 2 3 4 5 6 n terms: 3 – 9 27 –81 243 –729 a n

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Homework: Sequences: p664 #2, 4, 6, 12, 14, 16, 22, 26, 28, 30, 34 Series: pp 664–665 #38, 40, 42, 46, 48, 52, 55

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