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Introduction to Geometric Sequences and Series

Introduction to Geometric Sequences and Series. 23 May 2011. Investigation:. Find the next 3 terms of each sequence: {3, 6, 12, 24, …} {32, 16, 8, 4, …}. Geometric Sequences. Sequences that increase or decrease by multiplying the previous term by a fixed number

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Introduction to Geometric Sequences and Series

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  1. Introduction to Geometric Sequences and Series 23 May 2011

  2. Investigation: • Find the next 3 terms of each sequence: • {3, 6, 12, 24, …} • {32, 16, 8, 4, …}

  3. Geometric Sequences • Sequences that increase or decrease by multiplying the previous term by a fixed number • This fixed number is called r or the common ratio

  4. Finding the Common Ratio • Divide any term by its previous term • Find r, the common ratio: • {3, 9, 27, 81, …}

  5. Your Turn: • Find r, the common ratio: • {0.0625, 0.25, 1, 4, …} • {-252, 126, -63, 31.5, …}

  6. Arithmetic vs. Geometric Sequences Arithmetic Sequences • Increases by the common difference d • Addition or Subtraction • d = un – un–1 Geometric Sequences • Increases by the common ratio r • Multiplication or Division

  7. Your Turn: Classifying Sequences • Determine if each sequence is arithmetic, geometric, or neither: • {2, 7, 12, 17, 22, …} • {-6, -3.7, -1.4, 9, …} • {-1, -0.5, 0, 0.5, …} • {2, 6, 18, 54, 162, …}

  8. Recursive Form of a Geometric Sequence un = run–1 n ≥ 2 nth term n–1th term common ratio

  9. Example #1 • u1 = 2, u2 = 8 • Write the recursive formula • Find the next two terms

  10. Example #2 • u1 = 14, u2 = 39 • Write the recursive formula • Find the next two terms

  11. Your Turn: • For the following problems, write the recursive formula and find the next two terms: • u1 = 4, u2 = 4.25 • u1 = 90, u2 = -94.5

  12. Explicit Form of a Geometric Sequence common ratio un = u1rn–1 n ≥ 1 nth term 1st term

  13. Example #1 • u1 = 2, • Write the explicit formula • Find the next three terms • Find u12

  14. Example #2 • u1 = 6, u2 = 18 • Write the explicit formula • Find the next three terms • Find u12

  15. Your Turn: • For the following problems, write the explicit formula, find the next three terms, and find u12 • u1 = 5, r = -¼ • u1 = 5, u2 = -20 • u1 = 144, u2 = 72

  16. Partial Sum of a Geometric Sequence

  17. Example #1 • k = 9, u1 = -1.5, r = -½

  18. Example #2 • k = 6, u1 = 1, u2 = 5

  19. Example #3 • k = 8,

  20. Your Turn: • Find the partial sum: • k = 6, u1 = 5, r = ½ • k = 8, u1 = 9, r = ⅓ • k = 7, u1 = 3, u2 = 6 • k = 8, u1 = 24, u2 = 6

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