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Pre-Cal 20S January 21, 2009

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Understanding the relationship between arithmetic sequences and linear equations.

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Pre-Cal 20S January 21, 2009

  1. 1. The Story of Young Gauss ... or all about arithmetic series Photo Source: Karl Gauss (1777–1855)
  2. 2. Write the first five terms of the arithmetic sequence defined by: g(x) = –3x + 10 f(x) = 2x + 5 Sketch their graphs:
  3. 3. 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 the equation of a line. 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.
  4. 4. 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, ... Implicitly? Solution: a = 11 t = 11 + (51 - 1)(-6) 51 d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
  5. 5. Example: Find the 100th term t100 of the sequence 11, 5, -1, -7, ... tn = a + (n - 1)d Example: Find the 51st term t 51 of the sequence 11, 5, -1, -7, ...
  6. 6. What is the defining linear function that produces the following first four terms of an arithmetic sequence? 1, 3, 5, 7, . . . 2, 4, 6, 8, . . .
  7. 7. Find the number of terms in each of the following arithmetic sequences. 10, 15, 20, ... , 250 40, 38, 36, ... , -30

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