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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.7 Arithmetic and Geometric Sequences

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn About Arithmetic Sequences Geometric Sequences 5.7-2

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sequences A sequence is a list of numbers that are related to each other by a rule. The terms in a sequence are the numbers that form the sequence. 5.7-3

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Arithmetic Sequence An arithmetic sequence is a sequence in which each term after the first term differs from the preceding term by a constant amount. The common difference, d, is the amount by which each pair of successive terms differs. Example 1: 1, 5, 9, 13, 17,... d=4 5.7-4

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Write the first five terms of the arithmetic sequence with first term 9 and a common difference of –4. Solution The first five terms of the sequence are 9, 5, 1, –3, –7 5.7-5

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. General or nth Term of an Arithmetic Sequence For an arithmetic sequence with first term a 1 and common difference d, the general or nth term can be found using the following formula. a n = a 1 + (n – 1)d 5.7-6

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Determine the twelfth term of the arithmetic sequence whose first term is –5 and whose common difference is 3. Solution Replace: a 1 = –5, n = 12, d = 3 a n = a 1 + (n – 1)d a 12 = –5 + (12 – 1)3 = –5 + (11)3 = 28 5.7-7

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Write an expression for the general or nth term, a n, for the sequence 1, 6, 11, 16,… Solution Substitute: a 1 = 1, d = 5 a n = a 1 + (n – 1)d = 1 + (n – 1)5 = 1 + 5n – 5 = 5n – 4 5.7-8

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sum of the First n Terms of an Arithmetic Sequence The sum of the first n terms of an arithmetic sequence can be found with the following formula where a 1 represents the first term and a n represents the nth term. 5.7-9

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Determine the sum of the first 25 even natural numbers. Solution The sequence is 2, 4, 6, 8, 10, …, 50 Substitute a 1 = 2, a 25 = 50, n = 25 into the formula 5.7-10

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Geometric Sequences A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant. This constant is called the common ratio, r. r can be found by taking any term except the first and dividing it by the preceding term. 5.7-11

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 6: Write the first five terms of the geometric sequence whose first term, a 1, is 5 and whose common ratio, r, is 2. Solution The first five terms of the sequence are 5, 10, 20, 40, 80 5.7-12

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. General or nth Term of a Geometric Sequence For a geometric sequence with first term a 1 and common ratio r, the general or nth term can be found using the following formula. a n = a 1 r n–1 5.7-13

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: Determine the twelfth term of the geometric sequence whose first term is –4 and whose common ratio is 2. Solution Replace: a 1 = –4, n = 12, r = 2 a n = a 1 r n–1 a 12 = –4 2 12–1 = –4 2 11 = –4 2048 = –8192 5.7-14

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Write an expression for the general or nth term, a n, for the sequence 2, 6, 18, 54,… Solution Substitute: a 1 = 2, r = 3 a n = a 1 r n–1 = 2(3) n–1 5.7-15

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sum of the First n Terms of an Geometric Sequence The sum of the first n terms of an geometric sequence can be found with the following formula where a 1 represents the first term and r represents the common ratio. 5.7-16

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: Determine the sum of the first five terms in the geometric sequence whose first term is 4 and whose common ratio is 2. Solution Substitute a 1 = 4, r = 2, n = 5 into 5.7-17

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: Solution a 1 = 2, r = 2, n = 5 5.7-18

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