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Sequences




  Photo source chromaspiral
Find the next three terms in each sequence of numbers ...



                       4, 7, 10, 13,        ,       ,


     ...
4, 7, 10, 13,   ,   ,




      3n+1
4, 7, 10, 13,   ,   ,
3, 6, 12, 24,   ,   ,
Some Definitions


Sequence: An ordered list of numbers that follow a certain pattern (or rule).



Arithmetic Sequence:(i...
To Find The Common Difference

 d is the common difference        d = tn - t(n - 1)
 tn is an arbitrary term in the sequen...
Geometric Sequence: (i) Recursive Definition: An ordered list of numbers
      generated by continuously multiplying a val...
3, 6, 12, 24,   ,   ,
To Find The Common Ratio


  r is the common ratio
  tn is an arbitrary term in the sequence
  t(n - 1) is the term immedi...
Pre-Cal 40S Slides January 16,  2008
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Pre-Cal 40S Slides January 16, 2008

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Arithmetic and geometric sequences; implicit and recursive definitions.

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Pre-Cal 40S Slides January 16, 2008

  1. 1. Sequences Photo source chromaspiral
  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, , , 3n+1
  4. 4. 4, 7, 10, 13, , ,
  5. 5. 3, 6, 12, 24, , ,
  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) 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. 3, 6, 12, 24, , ,
  10. 10. 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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