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 About

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 in a sequence are the numbers that form the sequence.

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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

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Example 2:

Write 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, –7

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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:

Determine 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:Write 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:

Determine the sum of the first 25 even natural numbers.

125 2 50( ) 25 52

2 2 21300

6502

nn

n a as

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

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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:

Write 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:Determine 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:

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:Determine 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

5

14 1 (2)(1 )

1 1 2

n

n

a rs

r

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Example 9:

Solutiona1 = 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