Pre-Cal 40S Slides June 2, 2008

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Introduction to sequences.

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Pre-Cal 40S Slides June 2, 2008

  1. 1. Sequences Superman Sequence by flickr user nib.ipernity.com
  2. 2. Find the next three terms in each sequence of numbers ... 4, 7, 10, 13, , , 3, 6, 12, 24, , , 32, 16, 8, 4, , , 1, 1, 2, 3, 5, 8,13, , ,
  3. 3. 4, 7, 10, 13, 16 , 19 , 22
  4. 4. 3, 6, 12, 24, , ,
  5. 5. Write the implicit definition for this sequence. 32, 16, 8, 4, , ,
  6. 6. Some Definitions Sequence: An ordered list of numbers that follow a certain pattern (or rule). Arithmetic Sequence:(i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation. Common Difference (d):(i) The number that is repeatedly added to successive terms in an arithmetic sequence. (ii) From the implicit definition, d is the slope of the linear equation.
  7. 7. To Find The Common Difference d is the common difference d = tn - t(n - 1) tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence To Find the nth Term In an Arithmetic Sequence tn is the nth term tn = a + (n - 1)d a is the first term n is the quot;rankquot; of the nth term in the sequence d is the common difference Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ... Solution: a = 11 t51 = 11 + (51 - 1)(-6) Implicitly d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
  8. 8. Geometric Sequence: (i) Recursive Definition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by an exponential equation. Common Ratio (r): (i) The number that is repeatedly multiplied to successive terms in a geometic sequence. (ii) From the implicit definition, r is the base of the exponential function.
  9. 9. To Find The Common Ratio r is the common ratio tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence To Find the nth Term In a Geometic Sequence tn is the nth term a is the first term n is the quot;rankquot; of the nth term in the sequence r is the common ratio

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