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Chapter 12

Sequences and Series. Chapter 12. Section 12.1: Arithmetic Sequences and Series. Objectives: I can find the n th term and arithmetic means of an arithmetic sequence. I can find the sum of n terms of an arithmetic series. SEQUENCE. Definition : An arithmetic sequence.

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Chapter 12

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  1. Sequences and Series Chapter 12

  2. Section 12.1: Arithmetic Sequences and Series Objectives: • I can find the nth term and arithmetic means of an arithmetic sequence. • I can find the sum of n terms of an arithmetic series.

  3. SEQUENCE Definition: An arithmetic sequence A common difference

  4. Example 1: Find the next four terms in the arithmetic sequence: -5, -2, 1, … Recursive Formula Explicit Formula 4, 7, 10, 13, … C.D. = +3

  5. Example 2: Example 3: Find the first term in the arithmetic sequence for which: and . Find the 47th term in the arithmetic sequence: -4, -1, 2, 5, …

  6. Example 4: Write an arithmetic sequence that has five arithmetic means between 4.9 and 2.5. 4.9, _____, _____, _____, _____, _____, 2.5 4.5 4.1 3.7 3.3 2.9

  7. DIFFERENCE BETWEEN SEQUNECE AND SERIES Arithmetic Sequence Arithmetic Series -9, -3, 3, 9 -9 + -3 + 3 + 9 The symbol, , called: the nth partial sum.

  8. Example 5: Find the sum of the first 60 terms in the arithmetic series 9 + 14 + 19 + … + 304.

  9. Warm-Up Use the sequence to answer the questions 7.2, 6.6, 6, 5.4, … A. Write the Recursive and Explicit formulas B. Find the 30th Term C. Find the 18th Partial Sum

  10. Fill in Reference page for 12.1 • Recursive Arithmetic Sequence • Explicit Arithmetic Sequence • Arithmetic Series

  11. Section 12.2: Geometric Sequences and Series Objectives: • Find the nth term and geometric means of a geometric sequence. • Find the sum of n terms of a geometric sequence.

  12. Definition: A geometric sequence Example 1: Determine the common ratio and find the next three terms in each sequence: a. b. r – 1, -3r + 3, 9r – 9, … r = -.5 Common ratio, “r” r = -3

  13. Definition: The nth term of a geometric sequence Recursive Explicit

  14. Example 2: Find an approximation for the 23rd term in the sequence: 256, -179.2, 125.44, …

  15. G1 = 26000 G2 = year 1 G3 = year 2 G4 = year 3 G5 = year 4 Example 3: A new car costing $26,000 depreciates at a rate of 40% per year for four years. Find the value of the car at the end of four years. • NOTE: Geometric sequences can represent growth or decay.

  16. Example 4: Write a sequence that has two geometric means between 48 and -750. 48, _________, _________, -750 -120 300

  17. Definition: A geometric series Example 5: Find the sum of the first ten terms of the geometric series 16 – 48 + 144 – 432 + …

  18. Do homework TOGETHER!!!!

  19. Warm-Up 14, ____, ____, 112/27…. 28/3 56/9 • Find the two geometric means between the given numbers. • Write a recursive and Explicit formula for both. g4 = 14r3 112/27 = 14r3 r = 2/3 gn = 14(2/3)n-1 g1 = 14 gn = 2/3gn-1

  20. Warm-Up • Quick Check Self-Quiz (ungraded) • On the front desk • ONLY 5 minutes after the bell!!!!

  21. Fill in reference sheet for Geometric!

  22. Section 12.3: Infinite Sequences and Series Objectives: • Find the limit of the terms of an infinite sequence. • Find the sum of an infinite geometric series.

  23. Limit Notation: Example 1: • Estimate the limit of Bill Gates RULE!!!!!

  24. v Example 2: Find each limit a. b. 3 5 Example 1: Estimate the limi

  25. Example 2 continued… Find each limit. c. d.

  26. e.

  27. Definitions (Sum of an Infinite Series): ***Lets think about growing sequences for a minute*** Arithmetic Geometric Ratio: r > 1 Ratio: r = 1 Ratio: r < 1 ……… keeps growing forever….. …….keeps growing forever….. ……numbers stay the same, but it still keeps growing forever... ………numbers get smaller and smaller… ...hmmm interesting…….

  28. (Sum of an Infinite *Geometric* Series):

  29. Example 4: Find the sum of the series 21 – 3 +

  30. Example 5: Write as a fraction.

  31. Find the pattern….write the equation • 5, 7, 9, 11, …… • 6, 12, 24, 48….. • 7, 7/3, 7/9, 7/27, … • 1, 4, 9, 16,….. Always look for Arithmetic/Geometric FIRST Then, refer to “other” patterns

  32. Pattern Recognition 2 4 12 6 8 10 1 3 11 5 7 9 2 4 64 8 16 32 1 4 36 9 16 25 1 8 27 1 1*2 2 6 1*2*3 24 -1 1 -1 1

  33. 10 minutes for pattern recognition hw

  34. Warm-Up • Find your table’s Partial Sum (Sn), for the sequence defined by gn = 3(.2)n-1 • S1= • S2= • S3= • S4= • S5= • S6= • S7= • S8=

  35. Pattern Recognition 2 4 12 6 8 10 1 3 11 5 7 9 2 4 64 8 16 32 1 4 36 9 16 25 1 8 27 1 1*2 2 6 1*2*3 24 -1 1 -1 1

  36. Section 12.4: Convergent and Divergent Series Objectives: • Determine whether a series is convergent or divergent.

  37. Definitions: • Converge • Diverge Approaches something Keeps growing/decaying to infinity

  38. Example 1:Determine whether each arithmetic or geometric series is convergent or divergent. a. b. 2 + 4 + 8 + 16 + … c. 10 + 8.5 + 7 + 5.5 + … convergent R = -.5 geometric divergent R = 2 geometric Arithmetic…. divergent

  39. Ratio Test: NOTE about the ratio test, this test can only be used when all terms of the series are positive. r>1 divergent r<1 convergent r= 1 “No results”

  40. Example 2b. (do this one first)

  41. Definition: (found in section 12.5) The expression n! (n factorial) is defined as follows for n, an integer greater than zero. n! = n(n – 1)(n – 2)….1 Question: What is

  42. lim lim lim Example 2:Use the ratio test to determine whether each series is convergent or divergent. a. r<1, therefore convergent

  43. Example 2 continued…Use the ratio test to determine whether each series is convergent or divergent. c. d. Harmonic…

  44. Warm-Up Determine whether the following series converge/diverge: • 4/3, 4/9, 4/27……. • 3.2 + 3.84 + 4.608 + 5.5296…

  45. Patterns worksheet • n! = “n – factorial” • n(n-1)(n-2)(n-3)…..(1) ex: 4! = 4*3*2*1 = 24 2. Other cool patterns….. a. 1, 4, 9, 16, 25, …….. n2 b. 1, 2, 6, 24, 120, …. n! c. 1, 8, 27, 64, ….. n3 • 1. 5n • 2. (3/2)2n • 3. -3n + 10 • 4. 2n! • 5. an = an-1 + an-2 • 6. n3 • 7. n2 + 1 • 8. (n-1)! • 9 n/2n • 10. n/(n+1)

  46. Comparison Test: When n>1 Squish: A series converges if it is less than or equal to a similar, convergent series. Push: A series diverges if it is greater than or equal to a similar, divergent series.

  47. Push: A series diverges if it is greater than or equal to a similar, divergent series.

  48. Squish: A series converges if it is less than or equal to a similar, convergent series.

  49. Example 3:Use the comparison test to determine whether the following series are convergent or divergent: a. Divergent

  50. Example 3:Use the comparison test to determine whether the following series are convergent or divergent: b. Convergent

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