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GPS – Sequences and Series MA3A9. Students will use sequences and series a. Use and find recursive and explicit formulae for the terms of sequences. b. Recognize and use simple arithmetic and geometric sequences. c. Investigate limits of sequences. d. Use mathematical induction to find and prove formulae for sums of finite series. e. Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric series. f. Use summation notation to explore series. g. Determine geometric series and their limits.

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Chapter 14 Sequences and Series

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14-1 : Introduction to Sequences and Series Sequence – 1) an ordered list of numbers. 2) a function whose domain is the set of positive integers. Series – the sum of the numbers in a sequence. Finite Sequence – has a countable number of terms. Infinite Sequence – has an uncountable number of terms.

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14-1 : Introduction to Sequences and Series Describe the pattern and find the next three terms. 2,4,6,8,__,__,__ 5,2,-1,-4,__,__,__ 3,6,12,24,__,__,__ 1,4,9,16,__,__,__ 1,3/2,5/3,7/4,__,__,__

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14-1 : Introduction to Sequences and Series Recursive Formula – A formula for terms of a sequence that specifies each term as a function of the preceding term(s). Explicit Formula – A formula for terms of a sequence that specifies each term as a function of n (the number of the specified term)

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14-2 : Arithmetic, Geometric, and Other Sequences Discrete Function – A function whose domain is a set of disconnected values. Continuous Function – A function whose domain has no gaps or disconnected values. A sequence is a discrete function.

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14-2 : Arithmetic, Geometric, and Other Sequences Arithmetic Sequence – a sequence formed by adding the same number to each preceding term. d is the common difference (the number added to all preceding terms) Recursive Formula: a n =a n-1 +d Explicit Formula: a n =a 1 +d(n-1)

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14-2 : Arithmetic, Geometric, and Other Sequences Arithmetic Sequences 3,5,7,9,… Recursive formula: a 1 =3, a n =a n-1 +2 Explicit formula: a n =2n+1 5,2,-1,-4,… Recursive formula: Explicit formula: What would the graph of the terms of an arithmetic sequence look like?

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14-2 : Arithmetic, Geometric, and Other Sequences Sum of an Arithmetic Series S n =n / 2 (a 1 + a n )

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14-2 : Arithmetic, Geometric, and Other Sequences Geometric – a sequence formed by multiplying the same number to each preceding term. r is the common ratio (the number multiplied by all preceding terms) Recursive Formula: a n =r(a n-1 ) Explicit formula: a n =a 1 (r) n-1

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14-2 : Arithmetic, Geometric, and Other Sequences Geometric Sequences 3,6,12,24,… Recursive formula: a 1 =3, a n =2a n-1 Explicit formula: a n =3(2) n-1 4,-8,16,-32,… Recursive formula: Explicit formula: What would the graph of the terms of a geometric sequence look like?

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14-2 : Arithmetic, Geometric, and Other Sequences Sum of a Finite Geometric Series S n = a 1 (1 – r n ) / (1 – r ) Sum of an Infinite Geometric Series Abs value of r < 1 S n = a 1 / (1 – r )

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