Lesson 6 -- Aritmetic and Geometric Sequences and Series

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11.1,11.2 – Arithmetic Sequences and Series

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An introduction………… 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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Find the next four terms of –9, -2, 5, … Arithmetic Sequence 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms…… 12, 19, 26, 33

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Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k

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Vocabulary of Sequences (Universal)

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Given an arithmetic sequence with x 15 38 NA -3 X = 80

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-19 63 ?? x 6 353

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Try this one: 1.5 16 x NA 0.5

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9 x 633 NA 24 X = 27

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-6 29 20 NA x

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Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 4 5 NA x The two arithmetic means are –1 and 2, since –4, -1, 2 , 5 forms an arithmetic sequence

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Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 5 4 NA x The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4 , 4 forms an arithmetic sequence

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Find n for the series in which 5 x y 440 3 X = 16 Factor!

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The 4 th term of an arithmetic sequence is -9. The 8 th term is 21. Find the first term and then 9 th term. ____ _____ _____ -9 ____ ____ ____ 21 ____ 1 st 5th 21 = -9 + (5-1)d 21 = -9 +4d 7.5 = d A 4 = A1 + (4 -1)7.5 -9 = A1 + 3(7.5) -31.5 = A1 A9 = 21 + 7.5 = 28.5

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12.2 – 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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Vocabulary of Sequences (Universal)

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Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic

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1/2 x 9 NA 2/3 And S 9 S n

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Find two geometric means between –2 and 54 -2, ____, ____, 54 -2 54 4 NA x The two geometric means are 6 and -18, since –2, 6, -18 , 54 forms an geometric sequence

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-3, ____, ____, ____

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x 9 NA

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x 5 NA

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Insert one geometric mean between ¼ and 4 1/4 3 NA

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1/2 7 x 127/128

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Section 12.3 – 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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Find the sum, if possible:

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Find the sum, if possible:

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Find the sum, if possible:

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Find the sum, if possible:

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The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel? 50 40 32 32/5 40 32 32/5

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The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel? 100 75 225/4 100 75 225/4

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Sigma Notation

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UPPER BOUND (NUMBER) LOWER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE)

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Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3

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Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = ½

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Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric 19 + 18 + 16 + 12 + 4 -1 -2 -4 -8

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Rewrite the following using sigma notation: Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION: