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10.2 – Arithmetic Sequences and Series

10.2 – Arithmetic Sequences and Series. Arithmetic Series. Geometric Series. Sum of Terms. Sum of Terms. An introduction … describe the pattern. Arithmetic Sequences. Geometric Sequences. ADD To get next term. MULTIPLY To get next term. Find the next four terms of –9, -2, 5, ….

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10.2 – Arithmetic Sequences and Series

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  1. 10.2 – Arithmetic Sequences and Series

  2. Arithmetic Series Geometric Series Sum of Terms Sum of Terms An introduction … describe the pattern Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term

  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 terms……12, 19, 26, 33

  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

  5. Vocabulary of Sequences (Universal)

  6. Given an arithmetic sequence with x 38 15 NA -3 X = 80

  7. -19 353 ?? 63 x 6

  8. 1.5 x 16 NA 0.5 Try this one:

  9. 9 633 x NA 24 X = 27

  10. -6 20 29 NA x

  11. The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 5 4 NA x

  12. 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 Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 4 5 NA x

  13. Find n for the series in which 5 y x 440 3 Graph on positive window X = 16

  14. 10.3 – Geometric Sequences and Series

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

  16. Vocabulary of Sequences (Universal)

  17. Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic To find r, divide any term in the sequence by its preceding term. a2/a1 a3/a2

  18. 1/2 x 9 NA 2/3

  19. The two geometric means are 6 and -18, since –2, 6, -18, 54 forms an geometric sequence Find two geometric means between –2 and 54 -2, ____, ____, 54 -2 54 4 NA x

  20. x 9 NA

  21. x 5 NA

  22. *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1/4 3 NA

  23. 1/2 7 x

  24. Section 12.3 – Infinite Series

  25. 1, 4, 7, 10, 13, …. No Sum Infinite Arithmetic Finite Arithmetic 3, 7, 11, …, 51 Finite Geometric 1, 2, 4, …, 64 1, ±2, 4,± 8, … Infinite Geometric r > 1 r < -1 No Sum Infinite Geometric -1 < r < 1

  26. Find the sum, if possible: What? If possible? What are they talking about? If r is between -1 and 1!

  27. Find the sum, if possible:

  28. Find the sum, if possible:

  29. Find the sum, if possible:

  30. Find the sum, if possible:

  31. Converting repeating decimals to fractions • Write the repeating decimal 0.808080… as a fraction. • Write the repeating decimal 0.153153… as a fraction.

  32. 50 40 40 32 32 32/5 32/5 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?

  33. 100 100 75 75 225/4 225/4 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?

  34. Sigma Notation Section 12-5

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

  36. Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3

  37. Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = ½

  38. Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric 19 + 18 + 16 + 12 + 4 -1 -2 -4 -8

  39. Rewrite the following using sigma notation: Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION:

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