9.3 geometric sequences and series. objective to find specified terms and the common ratio in a...
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9.3 Geometric Sequences and Series
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Objective
• To find specified terms and the common ratio in a geometric sequence.
• To find the partial sum of a geometric series
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Geometric Sequences
• Consecutive terms of a geometric sequence have a common ratio.
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Definition of a Geometric Sequence
• A sequence is geometric if the ratios of consecutive terms are the same.
• The number r is the common ratio of the sequence.
32 4
1 2 3
, , ,..., 0aa a
r r r ra a a
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Example 1Examples of Geometric
Sequences• a). The sequence whose nth term is
• b). The sequence whose nth term is
• C) The sequence whose nth term is
2n
4(3 )n
1( )
3n
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• Notice that each of the geometric sequences has an nth term that is of the form where the common ratio is r.
• A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers.
nar
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The nth Term of a Geometric Sequence
• The nth term of a geometric sequence has the form where r is the common ratio of consecutive terms of the sequence.
11n
na a r
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• So, every geometric sequence can be written in the following form,
1 2 3 4
2 3 11 1 1 1 1
,. . .
, , , ,
n
n
a a a a a
a a r a r a r a r
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• If you know the nth term of a geometric sequence, you can find the (n+1)th term by multiplying by r. that is
1n na ra
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Example 2Finding the Terms of a Geometric Sequence
Write the first five terms of the geometric sequence whose first term is
and whose common ratio is r = 2.
3, 6, 12, 24, 48
1 3a
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Example 3Finding a Term of a
Geometric Sequence• Find the 15th term of the geometric sequence
whose first term is 20 and whose common ration is 1.05.
1415 20(1.05) 39.6a
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Example 4Finding a Term of a Geometric
Sequence• Find the 12th term of the geometric
sequence 5, 15, 45, . . .
1112
3
5(3) 885735
r
a
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• If you know any two terms of a geometric sequence, you can use that information to find a formula for the nth term of the sequence.
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Example 5Finding a Term of a
Geometric Sequence
• The fourth term of a geometric sequence is 125, and the 10th term is 125/64. Find the 14th term. (assume that the terms of the sequence are positive.)
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34 1
910 1
1
1314
125
125
64Two equations in two unknowns
Solve using substitution or elimination
1, 1000
21
1000( ) .122072
a a r
a a r
r a
a
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The Sum of a Finite Geometric Sequence
• The sum of the geometric sequence
with common ratio is given by
2 3 11 1 1 1 1, , , , na a r a r a r a r
1r
11 1
1
1
1
nni
ni
rS a r a
r
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Example 6Finding the Sum of a Finite
Geometric Sequence• Find the sum
121
1
4(0.3)i
i
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1
12
12 1
12
4, .3, 12
(1 )
(1 )
(1 (.3) )4
(1 .3)
5.71
a r n
rS a
r
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• When using the formula for the sum of a finite geometric sequence, be careful to check that the index begins at . If the index begins at , you must adjust the formula for the th partial sum.
4
0
4(0.3)i
i
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These are not the same, be careful of the indices
40 1 2 3 4
0
41 1 2 3 4
1
4(0.3) 4(0.3) 4(0.3) 4(0.3) 4(0.3) 4(0.3)
4(0.3) 4(0.3) 4(0.3) 4(0.3) 4(0.3)
i
i
i
i
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Geometric Series
• The summation of the terms of an infinite geometric sequence is called an infinite geometric series or geometric series.
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The sum of an Infinite Geometric Series
• If the infinite geometric series
has the sum
1,r 2 3 1
1 1 1 1 1, , , , ,...na a r a r a r a r
11
0 1i
i
aS a r
r
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Example 7Finding the Sums of an
Infinite Geometric Series
• Find the sums.
• a)
• b)
0
4(0.6)n
n
3 0.3 0.03 0.003 ...
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0
44(0.6) 10
1 .6
33 0.3 0.03 0.003 ...
1 .13 1
3.9 3
n
n
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ApplicationsCompound Interest
• A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance of this annuity at the end of 2 years?
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2424
2323
11
24
24
24
.0650(1 )
12.06
50(1 )12
.
.
.
.0650(1 )
12
1 (1.005)50(1 .005)
1 (1.005)
$1277.95
A
A
A
S
S