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Pre-Cal 40S June 4, 2009

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Arithmetic and geometric series.

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Pre-Cal 40S June 4, 2009

  1. 1. The Legacy of Karl Fredrich Gauss that is ... unstacking by flickr user mikelietz Zehner by flickr user threedots
  2. 2. Allan is one of 7 men and Brigit is one of 10 women who wish to be chosen for the show The Greatest Mathematician. From this group, 4 men and 4 women will be chosen. What is the probability that both Allan and Brigit will be among the 8 people chosen? Briefly explain your calculations.
  3. 3. Introduction to today's class by Mr. Green on YouTube ... a summary of almost everything in this unit ... Sequences and Series on YouTube http://youtube.com/watch?v=WjLSz-nNLBc
  4. 4. To Find the nth Term In an Arithmetic Sequence t is the nth term n t = a + (n - 1)d a is the first term n 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
  5. 5. 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
  6. 6. http://www.sigmaxi.org/amscionline/gauss-snippets.html The Story of Young Gauss ... Photo Source: Karl Gauss (1777–1855)
  7. 7. Series: The sum of numbers in a sequence to a particular term in a sequence. Example: denotes the sum of the first 5 terms. denotes the sum of the first n terms. Artithmetic Series: The sum of numbers in an arithmetic sequence given by is the sum to the nth term n is the quot;rankquot; of the nth term a is the first term in the sequence d is the common difference
  8. 8. (a) What is the sum of the integers from 1 to 5000? (b) What is the sum of all multiples of 7 between 1 & 5000? (c) What is the sum of all integers from 1 to 5000 inclusive that are not multiples of 7?
  9. 9. Sigma Notation: A shorthand way to write a series. Example: 4 ∑(2n - 3) means (2(1) -3) + (2(2) -3) + (2(3) -3) + (2(4) -3) n=1 = -1 + 1 + 3 + 5 =8 Σ is capital sigma (from the greek alphabet); means sum subscript n = 1 means quot;start with n = 1 and evaluate (2n - 3)quot; superscript 4 means keep evaluating (2n - 3) for successive integral values of n; stop when n = 4; then add all the terms (2n - 3) is the implicit definition of the sequence
  10. 10. Find the value of:
  11. 11. Series: The sum of numbers in a sequence to a particular term in a sequence. Example: denotes the sum of the first 5 terms. denotes the sum of the first n terms. Geometric Series: The sum of numbers in an geometric sequence given by or is the sum to the nth term n is the quot;rankquot; of the nth term a is the first term in the sequence d is the common difference

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