13 1 arithmetic and geometric sequences

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13 1 arithmetic and geometric sequences

  1. 1. Objectives: 1. Identify arithmetic and geometric sequences 2. Find formulas for the nth term
  2. 2. What is a Sequence? A set of numbers, called terms, arranged in a particular order. Two simplest types:  Arithmetic  Geometric
  3. 3. Arithmetic Sequences Difference between consecutive terms is constant.  Called the “common difference.” Examples: 2, 6, 10, 14, 18, … diff. = 4 17, 10, 3, -4, -11, -18, … diff. = -7  a, a+d, a+2d, a+3d, a+4d, … diff. = d
  4. 4. Geometric Sequences Ratio of consecutive terms is constant.  Called the “common ratio.” Examples: 1, 3, 9, 27, 81, … ratio= 3 64, -32, 16, -8, 4, … ratio = -1/2  a, ar, ar2, ar3, ar4, … ratio = r
  5. 5. You Try! Identify the type of sequence and the common difference or ratio. 5, 10, 20, 40, … 6, 1, -4, -9, …
  6. 6. Notation 1st term: t1, 2nd term: t2, nth term: tn Some sequences can be defined by rules or formulas. Ex: tn = n2 + 1 t1 = 12 + 1 = 2 t2 = 22 + 1 = 5, and so on
  7. 7. Arithmetic Formulas tn nth term = t0 + 0th term (work backwards to find) dn add the difference n times.
  8. 8. Geometric Formulas tn nth term = t0 0th term ∙ rn multiply by the ratio n times
  9. 9. Formal Definition “A function whose domain is the set of positive integers.” For example:  The sequence tn = 4n – 2 - can be thought of as  The function t(n) = 4n – 2 (where n is a + integer)
  10. 10. Graphing Sequences Write terms as ordered pairs and plot.  Ex: 1, 4, 7, 10, … has points (1, 1), (2, 4), (3, 7), (4, 10) Notice n (the term number) is the x! Arithmetic – points lie on a line Geometric – points lie on an exponential curve
  11. 11. Example 1: Find formula for nth term of 3, 5, 7… Sketch the graph.
  12. 12. Example 2: Find formula for nth term of 3, 4.5, 6.75… Sketch the graph.
  13. 13. You Try! Find formula for nth term of 15, 7, -1, -9, … 100, -50, 25, -12.5, …
  14. 14. Example 3: In a geometric sequence, t3 = 12 and t6 = 96. Find t11.
  15. 15. Example 4: In an arithmetic sequence t2 = 2 and t5 = 16. Find t10.
  16. 16. You Try! In a geometric sequence t2 = 2 and t5 = 16. Find t10.

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