Section 5.7 Arithmetic and Geometric Sequences

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 5.7

Arithmetic and

Geometric Sequences


What You Will Learn

Arithmetic Sequences

Geometric Sequences

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Sequences

A sequence is a list of numbers that are related to each other by a rule.The terms are the numbers that form the sequence.

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Arithmetic SequenceAn 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.To find the difference, simply subtract any term from the term that directly follows it.

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Example 2: An Arithmetic Sequence with a Negative DifferenceWrite the first five terms of the arithmetic sequence with first term 9 and a common difference of –4.

SolutionThe first five terms of the sequence are

9, 5, 1, –3, –75.7-5


General or nth Term of an Arithmetic SequenceFor an arithmetic sequence with first term a1 and common difference d, the general or nth term can be found using the following formula.

an = a1 + (n – 1)d

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Example 3: Determining the 12th Term of an Arithmetic SequenceDetermine the twelfth term of the arithmetic sequence whose first term is –5 and whose common difference is 3.

SolutionReplace: a1 = –5, n = 12, d = 3

an = a1 + (n – 1)da12 = –5 + (12 – 1)3

= –5 + (11)3= 28

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Example 4: Determining an Expression for the nth TermWrite an expression for the general or nth term, an, for the sequence

1, 6, 11, 16,…SolutionSubstitute: a1 = 1, d = 5

an = a1 + (n – 1)d= 1 + (n – 1)5= 1 + 5n – 5= 5n – 4

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Sum of the First n Terms of an Arithmetic SequenceThe sum of the first n terms of an arithmetic sequence can be found with the following formula where a1 represents the first term and an represents the nth term.

s

n

n(a1 a

n)

2

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Example 5: Determining the Sum of an Arithmetic SequenceDetermine the sum of the first 25 even natural numbers.

s

n

n(a1 a

n)

2

SolutionThe sequence is 2, 4, 6, 8, 10, …, 50Substitute a1 = 2, a25 = 50, n = 25 into the formula

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Example 5: Determining the Sum of an Arithmetic Sequence

s

25

25(2 50)

2

Solutiona1 = 2, a25 = 50, n = 25

s

25

25(52)

2

1300

2

s25650

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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.

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Example 6: The First Five Terms of a Geometric SequenceWrite the first five terms of the geometric sequence whose first term, a1, is 5 and whose common ratio, r, is 2.

SolutionThe first five terms of the sequence are

5, 10, 20, 40, 80

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General or nth Term of a Geometric SequenceFor a geometric sequence with first term a1 and common ratio r, the general or nth term can be found using the following formula.

an = a1r n–1

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Example 7: Determining the 12th Term of a Geometric SequenceDetermine the twelfth term of the geometric sequence whose first term is –4 and whose common ratio is 2.

SolutionReplace: a1 = –4, n = 12, r = 2

an = a1r n–1

a12 = –4 • 212–1

= –4 • 211 = –4 • 2048= –8192

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Example 8: Determining an Expression for the nth Term

Write an expression for the general or nth term, an, for the sequence

2, 6, 18, 54,…

SolutionSubstitute: a1 = 2, r = 3

an = a1r n–1

= 2(3)n–1

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Sum of the First n Terms of an Geometric SequenceThe sum of the first n terms of an geometric sequence can be found with the following formula where a1 represents the first term and r represents the common ratio.

s

n

a1(1 r n)

1 r, r 1

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Example 9: Determining the Sum of an Geometric SequenceDetermine the sum of the first five terms in the geometric sequence whose first term is 4 and whose common ratio is 2.

SolutionSubstitute a1 = 4, r = 2, n = 5 into

s

n

a1(1 r n)

1 r

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Example 5: Determining the Sum of an Arithmetic SequenceSolutiona1 = 2, r = 2, n = 5

s

5

4(1 32)

1

4 31 1

s5124

s

5

4 1 (2)5

1 2

124

1

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Documents

Section 5.7 Arithmetic and Geometric Sequences