7.3 - 1 geometric sequences a geometric sequence (or geometric progression) is a sequence in which...

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Geometric Sequences

A geometric sequence (or geometric progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero real number, called the common ratio. The sequence discussed in the last slide,

1,2,4,8,16, ,

is an example of a geometric sequence in which the first term is 1 and the commonratio is 2.

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Geometric Sequences

Notice that if we divide any term after the first term by the preceding term, we obtain the common ratio r = 2.

3 52 4

1 2 3 4

2 4 8 16; ; ; ;

1 2 4 82 2 2 2

a aa aa a a a

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Geometric Sequences

If the common ratio of a geometric sequence is r, then by the definition of a geometric sequence, 1 ,n

n

ar

a Common ratio r

for every positive integer n. Therefore, we find the common ratio by choosingany term except the first and dividing it by the preceding term.

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nth Term of a Geometric Sequence

In a geometric sequence with first term a1 and common ratio r, the nth term, an, is given by

11 .n

na a r

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Example 1 FINDING THE nth TERM OF A GEOMETRIC SEQUENCE

Use the formula for the nth term of a geometric sequence to answer the first question posed at the beginning of this section. How much will be earned on day 20 if daily wages follow the sequence 1, 2, 4, 8, 16,…cents?Solution

19 1920 1 1(2) 524,288 cents, or $5242.88a a r

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Example 3 FINDING TERMS OF A GEOMETRIC SEQUENCE

Find r and a1 for the geometric sequence with third term 20 and sixth term 160.

Use the formula for the nth term of a geometric sequence.

Solution

For n = 3, a3 = a1r2 = 20.

For n = 6, a6 = a1r5 = 160.

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Find r and a1 for the geometric sequence with third term 20 and sixth term 160.Solution

Since 21 1 2

2020, .a r a

r

Substitute this value for a1 in the second equation.

15 160a r

25 16

00

2r

r

Substitute.

320 160r Quotient rule for exponents

3 8r Divide by 20.

2r Take cube roots.

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Find r and a1 for the geometric sequence with third term 20 and sixth term 160.Solution

Since 21 20, and 2,a r r

21(2) 20a Substitute.

Divide by 4.

14 20a

1 5.a

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Geometric Series

A geometric series is the sum of the terms of a geometric sequence. In applications, it may be necessary to find the sum of the terms of such a sequence. For example, a scientist might want to know the total number of insects in four generations of the population discussed in Example 4.

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Sum of the First n Terms of a Geometric SequenceIf a geometric sequence has first term a1 and common ratio r, then the sumof the first n terms is given by

1(1 ), where 1.

1

n

n

a rS r

r

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Geometric Series

We can use a geometric series to find the total fruit fly population in Example 4 over the four-generation period. With n = 4, a1 =100, and r = 1.5,

which agrees with our previous result.

4

4

100(1 1.5 )

1 1.5S

100(1 5.0625)

.5

812.5

813 insects,

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Infinite Geometric Series

We extend our discussion of sums of sequences to include infinite geometric sequences such as

1 1 1 12,1, , , , , ,

2 4 8 16

with first term 2 and common ratio ½. Using the formula for Sn gives the following sequence of sums.

1 2 3 4 5

7 15 31 632, 3, , , , ,

2 4 8 16S S S S S S

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Example 7 SUMMING THE TERMS OF AN INFINITE GEOMETRIC SERIES

Evaluate 1 1 1

13 9 27

Solution

Use the formula for the sum of the first n terms of a geometric sequence to obtain

1 2 3 4

4 13 401, , , ,

3 9 27S S S S

and, in general,

13

1

1 1

.1

3

n

nSLet a1 = 1, r = 1/3.

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The table shows the value of (1/3)n for larger and larger values of n.

n 1 10 100 200

1

3

n1

3 51.69 10 481.94 10 963.76 10

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As n gets larger and larger, gets closer and closer to 0. That is,

1

3

n

1lim 0,

3

n

n

making it reasonable that

1 11(1 ) 1 3

lim lim .1 1 2 21 13 3 3

13 0

n

nnn

S

Hence, 1 1 1 3

1 .3 9 27 2

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This quotient, is called the sum of the

terms of an infinite geometric sequence.

The limit is often expressed as

1 ,

1

a

r

lim nn

S ii

1

or .S a

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Sum of The Terms of an Infinite Geometric SequenceThe sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where – 1 < r < 1, is given by

1 .1

aS

r

If │r│> 1, then the terms get larger and larger in absolute value, so there is no limit as n →∞ . Hence the terms of the sequence will not have a sum.

7.3 - 1 geometric sequences a geometric sequence (or geometric progression) is a sequence in which...

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