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**12.1 – Arithmetic Sequences and Series**

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An introduction………… Arithmetic Series Sum of Terms Geometric Series Sum of Terms Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term

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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 38 15 NA -3 X = 80

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-19 353 ?? 63 x 6

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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 5 4 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 4 5 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 y x 440 3 Graph on positive window X = 16

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**12.2 – Geometric Sequences and Series**

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

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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

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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*****

*** denotes trick question 1/4 3 NA

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1/2 7 x

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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 40 32 32 32/5 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? 100 100 75 75 225/4 225/4

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Sigma Notation

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

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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

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**Rewrite the following using sigma notation:**

Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION:

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