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Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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  • Slide 1
  • Geometric Sequences and Series
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  • Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms
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  • Geometric Sequence: sequence whose consecutive terms have a common ratio. Example: 3, 6, 12, 24, 48,... The terms have a common ratio of 2. The common ratio is the number r. To find the common ratio you use a n+1 a n
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  • Vocabulary of Sequences (Universal)
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  • Find the next two terms of 2, 6, 18, ___, ___ 6 2 vs. 18 6 not arithmetic 2, 6, 18, 54, 162 Find the next two terms of 80, 40, 20, ___, ___ 40 80 vs. 20 40 not arithmetic 80, 40, 20, 10, 5
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  • Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 2 vs. 9/2 3 not arithmetic Find the next two terms of -15, 30, -60, ___, ___ 30 -15 vs. -60 30 not arithmetic -15, 30, -60, 120, -240
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  • -3 anan 8 NA -2 Find the 8 th term if a 1 = -3 and r = -2.
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  • ?? anan 10 NA 3 Find the 10 th term if a 4 = 108 and r = 3. 4
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  • Write an equation for the n th term of the geometric sequence 3, 12, 48, 192, 3 4
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  • Geometric Mean: The terms between any two nonconsecutive terms of a geometric sequence. Ex. 2, 6, 18, 54, 162 6, 18, 54 are the Geometric Mean between 2 and 162
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  • Find two geometric means between 2 and 54 -2, ____, ____, 54 -2 54 4 NA r The two geometric means are 6 and -18, since 2, 6, -18, 54 forms a geometric sequence
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  • Geometric Series: An indicated sum of terms in a geometric sequence. Example: Geometric Sequence 3, 6, 12, 24, 48 VS Geometric Series 3 + 6 + 12 + 24 + 48
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  • Recall Vocabulary of Sequences (Universal)
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  • Application: Suppose you e-mail a joke to three friends on Monday. Each of those friends sends the joke to three of their friends on Tuesday. Each person who receives the joke on Tuesday sends it to three more people on Wednesday, and so on. Monday Tuesday # New people that receive jokeDay of WeekTotal # of people that received joke Monday Tuesday Wednesday 33 93 + 9 = 12 27 12 + 27 = 39
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  • 3 10 SnSn Find the sum of the first 10 terms of the geometric series 3 - 6 + 12 24+ -2
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  • In the book Roots, author Alex Haley traced his family history back many generations to the time one of his ancestors was brought to America from Africa. If you could trace your family back 15 generations, starting with your parents, how many ancestors would there be? 2 15 SnSn 2
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  • a1a1 8 39,360 NA 3
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  • 15,625 ?? SnSn -5
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  • Recall the properties of exponents. When multiplying like bases add exponents
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  • 15,625 ?? SnSn -5
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  • UPPER LIMIT (NUMBER) LOWER LIMIT (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) INDEX
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  • If the sequence is geometric (has a common ratio) you can use the S n formula 5+2 0 = 6 5 SnSn 2 5+2 5 = 37
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  • Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r =
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  • Infinite Series
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  • 1, 4, 7, 10, 13, . Infinite Arithmetic No Sum 3, 7, 11, , 51 Finite Arithmetic 1, 2, 4, , 64 Finite Geometric 1, 2, 4, 8, Infinite Geometric r > 1 r < -1 No Sum Infinite Geometric -1 < r < 1
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  • Find the sum, if possible:
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