1 / 74

Understanding Arithmetic and Geometric Sequences

Explore arithmetic and geometric sequences and their series, from finding terms to determining convergence. Learn formulas and examples for in-depth understanding.

ives
Download Presentation

Understanding Arithmetic and Geometric Sequences

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related