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*9.2 – Arithmetic Sequences and Series. An introduction Arithmetic Sequences ADD To get next...*

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9.2 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 Example: The nth Partial Sum The sum of the first n terms of an infinite sequence is called the nth partial sum. Slide 15 Example 6. Find the 150 th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, Slide 16 Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows? Slide 17 Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. So the total sales for the first 2o years is $1,625,000 Slide 18 9.3 Geometric Sequences and Series Slide 19 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 20 Vocabulary of Sequences (Universal) Slide 21 Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 2 vs. 9/2 3 not arithmetic Slide 22 1/2 x 9 NA 2/3 Slide 23 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 24 -3, ____, ____, ____ Slide 25 x 9 NA Slide 26 x 5 Slide 27 *** Insert one geometric mean between and 4*** *** denotes trick question 1/4 3 NA Slide 28 1/2 7 x Slide 29 Section 12.3 Infinite Series Slide 30 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 31 Find the sum, if possible: Slide 32 Slide 33 Slide 34 Slide 35 Slide 36 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 37 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 38 Sigma Notation Slide 39 UPPER BOUND (NUMBER) LOWER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) Slide 40 Slide 41 Slide 42 Slide 43 Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3 Slide 44 Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = Slide 45 Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric 19 + 18 + 16 + 12 + 4 -1 -2 -4 -8 Slide 46 Rewrite the following using sigma notation: Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION: