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12.1 – Arithmetic Sequences and Series. An introduction Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic

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12.1 Arithmetic Sequences and Series Slide 2 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 Slide 3 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 terms12, 19, 26, 33 Slide 4 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 Slide 5 Vocabulary of Sequences (Universal) Slide 6 Given an arithmetic sequence with x 15 38 NA -3 X = 80 Slide 7 -19 63 ?? x 6 353 Slide 8 Try this one: 1.5 16 x NA 0.5 Slide 9 9 x 633 NA 24 X = 27 Slide 10 -6 29 20 NA x Slide 11 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 Slide 12 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 Slide 13 Find n for the series in which 5 x y 440 3 X = 16 Graph on positive window Slide 14 12.2 Geometric Sequences and Series Slide 15 Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms Slide 16 Vocabulary of Sequences (Universal) Slide 17 Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 2 vs. 9/2 3 not arithmetic Slide 18 1/2 x 9 NA 2/3 Slide 19 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 Slide 20 -3, ____, ____, ____ Slide 21 x 9 NA Slide 22 x 5 Slide 23 *** Insert one geometric mean between and 4*** *** denotes trick question 1/4 3 NA Slide 24 1/2 7 x Slide 25 Section 12.3 Infinite Series Slide 26 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 Slide 27 Find the sum, if possible: Slide 28 Slide 29 Slide 30 Slide 31 Slide 32 The Bouncing Ball Problem Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of its height, and continues this pattern until it stops. How far does the ball travel? 50 40 32 32/5 40 32 32/5 Slide 33 The Bouncing Ball Problem Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of its height, and continues this pattern until it stops. How far does the ball travel? 100 75 225/4 100 75 225/4 Slide 34 Sigma Notation Slide 35 UPPER BOUND (NUMBER) LOWER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) Slide 36 Slide 37 Slide 38 Slide 39 Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3 Slide 40 Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = Slide 41 Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric 19 + 18 + 16 + 12 + 4 -1 -2 -4 -8 Slide 42 Rewrite the following using sigma notation: Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION:

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