mes96990_ch15_895-906.qxd
11/9/10
8:25 AM
Page 906
BIA—
906
Chapter 15 Sequences and Series
73) Beginning the first week of June, Noor will begin to
deposit money in her bank. She will deposit $1 the first
week, $2 the second week, $3 the third week, $4 the
fourth week, and she will continue to deposit money in
this way for 24 weeks. How much money will she have
saved after 24 weeks?
78) A theater has 23 rows. The first row contains 10 seats, the
next row has 12 seats, the next row has 14 seats, and so
on. How many seats are in the last row? How many seats
are in the theater?
74) Refer to Exercise 73. If Tracy deposits money weekly but
deposits $1 then $3 then $5, etc., how much will Tracy
have saved after 24 weeks?
75) A stack of logs has 12 logs in the bottom row (the first
row), 11 logs in the second row, 10 logs in the third row,
and so on, until the last row contains one log.
a) How many logs are in the eighth row?
b) How many logs are in the stack?
VIDEO
76) A landscaper plans to put a pyramid design in a brick
patio so that the bottom row of the pyramid contains 9
bricks and every row above it contains two fewer bricks.
How many bricks does she need to make the design?
79) The main floor of a concert hall seats 860 people. The
first row contains 24 seats, and the last row contains
62 seats. If each row has 2 more seats than the previous
row, how many rows of seats are on the main floor of the
concert hall?
77) A lecture hall has 14 rows. The first row has 12 seats, and
each row after that has 2 more seats than the previous
row. How many seats are in the last row? How many seats
are in the lecture hall?
80) A child builds a tower with blocks so that the bottom row
contains 9 blocks and the top row contains 1 block. If he
uses 45 blocks, how many rows are in the tower?
Section 15.3 Geometric Sequences and Series
Objectives
1. Define Geometric
Sequence and
Common Ratio
2. Find the Common
Ratio for a Geometric
Sequence
3. Find the Terms of a
Geometric Sequence
4. Find the General
Term of a Geometric
Sequence
5. Find a Specified Term
of a Geometric
Sequence
6. Solve an Applied
Problem Involving a
Geometric Sequence
7. Find the Sum of
Terms of a Geometric
Sequence
8. Find the Sum of
Terms of an Infinite
Geometric Sequence
9. Solve an Applied
Problem Involving an
Infinite Series
10. Distinguish Between
an Arithmetic and a
Geometric Sequence
1. Define Geometric Sequence and Common Ratio
In Section 15.2, we learned that a sequence such as
5, 9, 13, 17, 21, . . .
is an arithmetic sequence because each term differs from the previous term by a constant
amount, called the common difference, d. In this case, d 4.
The terms of the sequence
3, 6, 12, 24, 48, . . .
do not differ by a constant amount, but each term after the first is obtained by multiplying
the preceding term by 2. Such a sequence is called a geometric sequence.
Definition
A geometric sequence is a sequence in which each term after the first is obtained by multiplying
the preceding term by a constant, r. r is called the common ratio.
Note
A geometric sequence is also called a geometric progression.
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 907
BIA—
Section 15.3 Geometric Sequences and Series
907
In the sequence 3, 6, 12, 24, 48, . . . the common ratio, r, is 2. We can find the value of
r by dividing any term after the first by the preceding term. For example,
r⫽
6
12
24
48
⫽
⫽
⫽
⫽2
3
6
12
24
2. Find the Common Ratio for a Geometric Sequence
Example 1
Find the common ratio, r, for each geometric sequence.
a) 5, 15, 45, 135, . . .
b)
3
12, 6, 3, , . . .
2
Solution
a) To find r, choose any term and divide it by the term preceding it: r ⫽
15
⫽ 3.
5
It is important to realize that dividing any term by the term immediately before it will
give the same result.
1
3
b) Choose any term and divide by the term preceding it: r ⫽ ⫽ .
6
2
You Try 1
Find the common ratio, r, for each geometric sequence.
a)
2, 12, 72, 432, . . .
5 5
b) 15, 5, , , . . .
3 9
3. Find the Terms of a Geometric Sequence
Example 2
Write the first five terms of the geometric sequence with first term 10 and common ratio 2.
Solution
Each term after the first is obtained by multiplying by 2.
a1
a2
a3
a4
a5
⫽ 10
⫽ 10(2)
⫽ 20(2)
⫽ 40(2)
⫽ 80(2)
⫽ 20
⫽ 40
⫽ 80
⫽ 160
The first five terms of the sequence are 10, 20, 40, 80, 160.
You Try 2
Write the first five terms of the geometric sequence with first term 2 and common ratio 2.
■
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 908
BIA—
908
Chapter 15 Sequences and Series
4. Find the General Term of a Geometric Sequence
Example 2 suggests a pattern that enables us to find a formula for the general term, an, of
a geometric sequence. The common ratio, r, is 2. The first five terms of the sequence are
10
20
40
80
160
a1 ⫽ 10
a2 ⫽ 10 ⴢ 2
a2 ⫽ a1 ⴢ r
a3 ⫽ 10 ⴢ 4
a3 ⫽ a1 ⴢ r2
a4 ⫽ 10 ⴢ 8
a4 ⫽ a1 ⴢ r3
a5 ⫽ 10 ⴢ 16
a5 ⫽ a1 ⴢ r4
The exponent on r is one less than the term number.
This pattern applies for any geometric sequence. Therefore, the general term, an, of a
geometric sequence is given by an ⫽ a1r n⫺1.
Definition
General Term of a Geometric Sequence: The general term of a geometric sequence with first
term a1 and common ratio r is given by
an ⫽ a1r n⫺1
Example 3
Find the general term, an, and the eighth term of the geometric sequence 3, 6, 12,
24, 48, . . . .
Solution
a1 ⫽ 3 and r ⫽
6
⫽ 2. Substitute these values into an ⫽ a1r n⫺1.
3
an ⫽ a1r n⫺1
an ⫽ 3(2) n⫺1
Let a1 ⫽ 3 and r ⫽ 2.
The general term, an ⫽ 3(2)
, is in simplest form.
To find the eighth term, a8, substitute 8 for n and simplify.
n⫺1
an ⫽ 3(2) n⫺1
a8 ⫽ 3(2) 8⫺1
a8 ⫽ 3(2) 7
a8 ⫽ 3(128)
a8 ⫽ 384
Let n ⫽ 8.
Subtract.
27 ⫽ 128
Multiply.
The eighth term is 384.
Remember the order of operations when evaluating 3(2) 7. We evaluate exponents before we do
multiplication to find 27 ⫽ 128 before multiplying by 3.
You Try 3
Find the general term, an, and the fifth term of the geometric sequence 2, 6, 18, . . . .
■
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 909
BIA—
Section 15.3 Geometric Sequences and Series
909
5. Find a Specified Term of a Geometric Sequence
If we know a1 and r for a geometric sequence, we can find any term using an ⫽ a1r n⫺1.
Example 4
4
Find the sixth term of the geometric sequence 12, ⫺4, , . . . .
3
Solution
a1 ⫽ 12 and r ⫽ ⫺
4
1
⫽ ⫺ . Find a6 using an ⫽ a1r n⫺1.
12
3
an ⫽ a1rn⫺1
1 6⫺1
a6 ⫽ (12)a⫺ b
3
1 5
a6 ⫽ 12a⫺ b
3
1
a6 ⫽ 12a⫺
b
243
4
a6 ⫽ ⫺
81
The sixth term is ⫺
1
Let n ⫽ 6, a1 ⫽ 12, and r ⫽ ⫺ .
3
4
.
81
■
You Try 4
Find the seventh term of the geometric sequence ⫺50, 25, ⫺
25
,....
2
6. Solve an Applied Problem Involving a Geometric Sequence
Example 5
A pickup truck purchased for $24,000 (This is its value at the beginning of year 1.) depreciates 25% each year. That is, its value each year is 75% of its value the previous year.
a) Find the general term, an, of the geometric sequence that models the value of the
truck at the beginning of each year.
b) What is the pickup truck worth at the beginning of the fourth year?
Solution
a) To find the value of the pickup each year, we multiply how much it was worth the
previous year by 0.75. Therefore, the value of the truck each year can be modeled by
a geometric sequence.
a1 ⫽ 24,000 since this is the value at the beginning of year 1. r ⫽ 0.75 since we will
multiply the value each year by 0.75 to find the value the next year.
Substitute a1 ⫽ 24,000 and r ⫽ 0.75 into an ⫽ a1r n⫺1 to find the general term.
an ⫽ a1r n⫺1
an ⫽ 24,000(0.75) n⫺1
Let a1 ⫽ 24,000 and r ⫽ 0.75.
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 910
BIA—
910
Chapter 15 Sequences and Series
b) To find the value of the pickup truck at the beginning of the fourth year, use an from
a) and let n ⫽ 4.
an ⫽ 24,000(0.75) n⫺1
a4 ⫽ 24,000(0.75) 4⫺1
a4 ⫽ 24,000(0.75) 3
a4 ⫽ 10,125
Let n ⫽ 4.
The truck is worth $10,125 at the beginning of the fourth year.
■
You Try 5
A minivan purchased for $27,000 depreciates 30% each year. That is, its value each year is 70%
of its value the previous year.
a) Find the general term, an, of the geometric sequence that models the value of the minivan at
the beginning of each year.
b)
What is the minivan worth at the beginning of the third year?
Geometric Series
7. Find the Sum of Terms of a Geometric Sequence
A geometric series is a sum of terms of a geometric sequence.
Just as we can use a formula to find the sum of the first n terms of an arithmetic
sequence, there is a formula to find the sum of the first n terms of a geometric sequence.
Let Sn represent the sum of the first n terms of a geometric sequence. Then,
Sn ⫽ a1 ⫹ a1r ⫹ a1r2 ⫹ a1r3 ⫹ # # # ⫹ a1rn⫺1
Multiply both sides of the equation by r.
rSn ⫽ a1r ⫹ a1r2 ⫹ a1r3 ⫹ a1r4 ⫹ . . . ⫹ a1rn
Subtract rSn from Sn.
Sn ⫺ rSn ⫽ (a1 ⫺ a1r) ⫹ (a1r ⫺ a1r2 ) ⫹ (a1r2 ⫺ a1r3 ) ⫹ (a1r3 ⫺ a1r4 ) ⫹ . . .
⫹ (a1rn⫺1 ⫺ a1rn )
Regrouping the right-hand side gives us
Sn ⫺ rSn ⫽ a1 ⫹ (a1r ⫺ a1r) ⫹ (a1r2 ⫺ a1r2 ) ⫹ (a1r3 ⫺ a1r3 ) ⫹ . . .
⫹ (a1rn⫺1 ⫺ a1rn⫺1 ) ⫺ a1rn
The differences in parentheses equal zero, and we get Sn ⫺ rSn ⫽ a1 ⫺ a1rn.
Factor out Sn on the left-hand side and a1 on the right-hand side.
Sn (1 ⫺ r) ⫽ a1 (1 ⫺ rn )
a1 (1 ⫺ rn )
Sn ⫽
1⫺r
Divide by (1 ⫺ r).
Definition
Sum of the First n Terms of a Geometric Sequence: The sum of the first n terms, Sn, of a
geometric sequence with first term a1 and common ratio r is given by
Sn ⫽
where r ⫽ 1.
a1 (1 ⫺ r n )
1⫺r
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 911
BIA—
Section 15.3 Geometric Sequences and Series
911
Example 6
Find the sum of the first four terms of the geometric sequence with first term 2 and
common ratio 5.
Solution
a1 ⫽ 2, r ⫽ 5, and n ⫽ 4. We are asked to find S4, the sum of the first four terms of the
geometric sequence. Use the formula
Sn ⫽
S4 ⫽
a1 (1 ⫺ rn )
1⫺r
2[1 ⫺ 152 4 ]
1⫺5
2(1 ⫺ 625)
S4 ⫽
⫺4
2(⫺624)
S4 ⫽
⫺4
S4 ⫽ 312
Let n ⫽ 4, r ⫽ 5, and a1 ⫽ 2.
54 ⫽ 625
We will verify that this result is the same as the result we would obtain by finding the
first four terms of the sequence and then finding their sum.
a1 ⫽ 2
a2 ⫽ 2 ⴢ 5 ⫽ 10
a3 ⫽ 10 ⴢ 5 ⫽ 50
a4 ⫽ 50 ⴢ 5 ⫽ 250
The terms are 2, 10, 50, and 250. Their sum is 2 ⫹ 10 ⫹ 50 ⫹ 250 ⫽ 312.
■
You Try 6
Find the sum of the first five terms of the geometric sequence with first term 3 and common
ratio 4.
n
Using summation notation, a a ⴢ bi (where a and b are constants) represents a geometric
i⫽1
series or the sum of the first n terms of a geometric sequence. Furthermore, the first term
is found by substituting 1 for i (so that the first term is ab) and the common ratio is b. We
n
can evaluate a a ⴢ bi using the formula for Sn.
i⫽1
Example 7
5
Evaluate a 6(2) i.
i⫽1
Solution
Use the formula Sn ⫽
r ⫽ 2, and n ⫽ 5.
a1 (1 ⫺ rn )
to find the sum. If we let i ⫽ 1, we obtain a1 ⫽ 12,
1⫺r
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 912
BIA—
912
Chapter 15 Sequences and Series
Substitute these values into the formula for Sn.
Sn ⫽
S5 ⫽
a1 (1 ⫺ rn )
1⫺r
1211 ⫺ 25 2
Let n ⫽ 5, a1 ⫽ 12, and r ⫽ 2.
1⫺2
12(1 ⫺ 32)
S5 ⫽
⫺1
S5 ⫽ ⫺12(⫺31)
S5 ⫽ 372
25 ⫽ 32
5
i
a 6 ⴢ 2 ⫽ 372
■
i⫽1
You Try 7
4
Evaluate a 3(5) i.
i⫽1
8. Find the Sum of Terms of an Infinite Geometric Sequence
n
1
2
3
4
5
6
1 n
rn ⴝ a b
2
1
⫽ 0.5
2
1
⫽ 0.25
4
1
⫽ 0.125
8
1
⫽ 0.0625
16
1
⫽ 0.03125
32
1
⫽ 0.015625
64
o
15
1
⬇ 0.0000305
32,768
Until now, we have considered only the sum of the first n terms of a geometric sequence.
That is, we have discussed the sum of a finite series. Is it possible to find the sum of an
infinite series?
1
Consider a geometric series with common ratio r ⫽ . What happens to the value of
2
n
1
a b , or rn, as n gets larger?
2
1 n
We will make a table of values containing n and a b .
2
At the left you can see that as the value of n gets larger, the value of rn gets smaller. In fact,
the value of rn gets closer and closer to zero.
We say that as n approaches infinity, rn approaches zero.
How does this affect the formula for the sum of the first n terms of a geometric sequence?
a1 (1 ⫺ rn )
.
The formula is Sn ⫽
1⫺r
a1 (1 ⫺ 0)
a1
⫽
.
If n approaches infinity and rn approaches zero, we get S ⫽
1⫺r
1⫺r
This formula will hold for 冟r 冟 ⬍ 1.
Definition
Sum of the Terms of an Infinite Geometric Sequence: The sum of the terms, S, of an infinite
geometric sequence with first term a1 and common ratio r, where 冟 r 冟 ⬍ 1, is given by
S⫽
If 冟 r 冟 ⱖ 1, then the sum does not exist.
a1
.
1⫺r
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 913
BIA—
Section 15.3 Geometric Sequences and Series
913
Example 8
8 16
Find the sum of the terms of the infinite geometric sequence 6, 4, , , . . . .
3 9
Solution
We will use the formula S ⫽
a1
, so we must identify a1 and find r. a1 ⫽ 6.
1⫺r
r⫽
4
2
⫽
6
3
2
2
Since 冟r 冟 ⫽ ` ` ⬍ 1, the sum exists. Substitute a1 ⫽ 6 and r ⫽ into the formula.
3
3
S⫽
a1
⫽
1⫺r
6
1⫺
2
3
⫽
6
⫽ 6 ⴢ 3 ⫽ 18
1
3
The sum is 18.
■
You Try 8
3 9 27
Find the sum of the terms of the infinite geometric sequence 1, , ,
,....
5 25 125
Remember, if 冟r 冟 ⱖ 1 then the sum does not exist!
9. Solve an Applied Problem Involving an Infinite Series
Example 9
Each time a certain pendulum swings, it travels 80% of the distance
it traveled on the previous swing. If it travels 2 ft on its first swing,
find the total distance the pendulum travels before coming to rest.
Solution
The geometric series that models this problem is
2 ⫹ 1.6 ⫹ 1.28 ⫹ . . .
where
2 ⫽ number of feet traveled on the first swing
2(0.80) ⫽ 1.6 ⫽ number of feet traveled on the second swing
1.6(0.80) ⫽ 1.28 ⫽ number of feet traveled on the third swing
etc.
a1
We can use the formula S ⫽
with a1 ⫽ 2 and r ⫽ 0.80 to find the total distance
1⫺r
the pendulum travels before coming to rest.
S⫽
a1
2
2
⫽
⫽
⫽ 10
1⫺r
1 ⫺ 0.80
0.20
The pendulum travels 10 ft before coming to rest.
■
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 914
BIA—
914
Chapter 15 Sequences and Series
You Try 9
Each time a certain pendulum swings, it travels 90% of the distance it traveled on the previous
swing. If it travels 20 in. on its first swing, find the total distance the pendulum travels before
coming to rest.
10. Distinguish Between an Arithmetic and a Geometric Sequence
Example 10
Determine whether each sequence is arithmetic or geometric. Then find the sum of the
first six terms of each sequence.
a)
⫺8, ⫺4, ⫺2, ⫺1, . . .
b)
⫺9, ⫺4, 1, 6, . . .
Solution
a) If every term of the sequence is obtained by adding the same constant to the previous
term, then the sequence is arithmetic. (The sequence has a common difference, d.)
If every term is obtained by multiplying the previous term by the same constant,
then the sequence is geometric. (The sequence has a common ratio, r.)
By inspection we can see that the terms of the sequence
⫺8, ⫺4, ⫺2, ⫺1, . . .
are not obtained by adding the same amount to each term. For example,
⫺4 ⫽ ⫺8 ⫹ 4, but ⫺2 ⫽ ⫺4 ⫹ 2.
Is there a common ratio?
1
⫺4
⫽ ,
⫺8
2
⫺2
1
⫽ ,
⫺4
2
⫺1
1
⫽
⫺2
2
1
Yes, r ⫽ . The sequence is geometric.
2
a1 (1 ⫺ r n )
1
Use Sn ⫽
with a1 ⫽ ⫺8, r ⫽ , and n ⫽ 6 to find S6, the sum of the
1⫺r
2
first six terms of this geometric sequence.
Sn ⫽
S6 ⫽
S6 ⫽
S6 ⫽
S6 ⫽
a1 (1 ⫺ rn )
1⫺r
1 6
⫺8 c 1 ⫺ a b d
2
1
1⫺a b
2
1
⫺8 a1 ⫺ b
64
1
2
63
⫺8 a b
64
1
2
63
⫺
8
63
63
⫽⫺ ⴢ2⫽⫺
1
8
4
2
1
Let a1 ⫽ ⫺8, r ⫽ , and n ⫽ 6.
2
1 6
1
a b ⫽
2
64
Subtract.
The sum of the first six terms of the geometric sequence ⫺8, ⫺4, ⫺2, ⫺1, . . . is ⫺
63
.
4
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 915
BIA—
Section 15.3 Geometric Sequences and Series
915
b) Each term in the sequence ⫺9, ⫺4, 1, 6, is obtained by adding 5 to the previous
term. This is an arithmetic sequence with a1 ⫽ ⫺9 and common difference d ⫽ 5.
Since we know a1, d, and n (n ⫽ 6), we will use the formula
n
Sn ⫽ [2a1 ⫹ (n ⫺ 1)d]
2
to find S6, the sum of the first six terms of this arithmetic sequence.
6
S6 ⫽ [2(⫺9) ⫹ (6 ⫺ 1)5]
2
S6 ⫽ 3[⫺18 ⫹ (5)(5)]
S6 ⫽ 3[⫺18 ⫹ 25] ⫽ 3(7) ⫽ 21
Let a1 ⫽ ⫺9, d ⫽ 5, and n ⫽ 6.
The sum of the first six terms of the arithmetic sequence ⫺9, ⫺4, 1, 6, . . . is 21.
■
You Try 10
Determine whether each sequence is arithmetic or geometric. Then, find the sum of the first
seven terms of each sequence.
a)
25, 22, 19, 16, . . .
b)
1 1 2 4
, , , ,...
6 3 3 3
Answers to You Try Exercises
a) 6
5)
a) an ⫽ 27,000(0.70) n⫺1
9)
b)
1
3
1)
200 in.
2)
10)
2, 4, 8, 16, 32
3) an ⫽ 2(3) n⫺1; a5 ⫽ 162
b) $13,230
6) S5 ⫽ 1023
7)
25
32
5
2
4) a7 ⫽ ⫺
2340
8)
127
a) arithmetic; S7 ⫽ 112 b) geometric; S7 ⫽
6
15.3 Exercises
Mixed Exercises: Objectives 1 and 2
13) a1 ⫽ 72, r ⫽
1) What is the difference between an arithmetic and a
geometric series?
2) Give an example of a geometric sequence.
Find the common ratio, r, for each geometric sequence.
3) 1, 2, 4, 8, . . .
1
5) 9, 3, 1, , . . .
3
1 1 1
7) ⫺2, , ⫺ , , . . .
2 8 32
4) 3, 12, 48, 192, . . .
6) 8, 4, 2, 1, . . .
8) 2, ⫺6, 18, ⫺54, . . .
Objective 3: Find the Terms of a Geometric Sequence
Write the first five terms of the geometric sequence with the
given first term and common ratio.
VIDEO
9) a1 ⫽ 2, r ⫽ 5
10) a1 ⫽ 3, r ⫽ 2
1
11) a1 ⫽ , r ⫽ ⫺2
4
12) a1 ⫽ 250, r ⫽
1
5
2
3
14) a1 ⫽ ⫺20, r ⫽ ⫺
3
2
Mixed Exercises: Objectives 4 and 5
Find the general term, an, for each geometric sequence. Then,
find the indicated term.
15) a1 ⫽ 4, r ⫽ 7; a3
16) a1 ⫽ 3, r ⫽ 8; a3
17) a1 ⫽ ⫺1, r ⫽ 3; a5
1
18) a1 ⫽ ⫺5, r ⫽ ⫺ ; a4
3
1
19) a1 ⫽ 2, r ⫽ ; a4
5
1
3
21) a1 ⫽ ⫺ , r ⫽ ⫺ ; a4
2
2
20) a1 ⫽ 7, r ⫽ 3; a5
3
22) a1 ⫽ , r ⫽ 2; a6
5
Find the general term of each geometric sequence.
23) 5, 10, 20, 40, . . .
24) 4, 12, 36, 108, . . .
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 916
BIA—
916
VIDEO
Chapter 15 Sequences and Series
3
3
3
25) ⫺3, ⫺ , ⫺ , ⫺
,...
5 25 125
26) ⫺1, 4, ⫺16, 64, . . .
27) 3, ⫺6, 12, ⫺24, . . .
2 2 2
28) 2, , , , . . .
3 9 27
29)
46) A luxury car purchased for $64,000 depreciates 15% each
year.
1 1 1 1
, , ,
,...
3 12 48 192
1
3
9
27
30) ⫺ , ⫺ , ⫺ , ⫺ , . . .
5 10 20 40
Find the indicated term of each geometric sequence.
31) 1, 2, 4, 8, . . .; a12
32) 1, 3, 9, 27, . . .; a10
33) 27, ⫺9, 3, ⫺1, . . .; a8
34) ⫺
1
1
1
, ⫺ , ⫺ , ⫺1, . . .; a7
125 25 5
35) ⫺
1
1
1
1
, ⫺ , ⫺ , ⫺ , . . .; a12
64 32 16 8
36) ⫺5, 10, ⫺20, 40, . . .; a8
Objective 10: Distinguish Between an Arithmetic
and a Geometric Sequence
Determine whether each sequence is arithmetic or geometric.
Then, find the general term, an, of the sequence.
VIDEO
a) Find the general term, an, of the geometric sequence
that models the value of the car at the beginning of
each year.
b) How much is the luxury car worth at the beginning of
the fourth year?
47) A company’s advertising budget is currently $500,000 per
year. For the next several years, the company will cut the
budget by 10% per year.
38) ⫺1, ⫺3, ⫺9, ⫺27, ⫺81, . . .
a) Find the general term, an, of the geometric sequence
that models the company’s advertising budget for each
of the next several years.
39) ⫺2, 6, ⫺18, 54, ⫺162, . . .
b) What is the advertising budget 3 years from now?
37) 15, 24, 33, 42, 51, . . .
40) 8, 3, ⫺2, ⫺7, ⫺12, . . .
41)
1 1 1 1 1
, , , ,
,...
9 18 36 72 144
42) 11, 22, 44, 88, 176, . . .
43) ⫺31, ⫺24, ⫺17, ⫺10, ⫺3, . . .
44)
3
5
7
, 2, , 3, , . . .
2
2
2
Objective 6: Solve an Applied Problem Involving
a Geometric Sequence
Solve each application.
48) In January 2011, approximately 1000 customers at a
grocery store used the self-checkout lane. The owners
predict that number will increase by 20% per month for
the next year.
a) Find the general term, an, of the geometric sequence
that models the number of customers expected to use
the self-checkout lane each month for the next year.
b) Predict how many people will use the self-checkout
lane in September 2011. Round to the nearest whole
number.
49) A home purchased for $160,000 increases in value by
4% per year.
45) A sports car purchased for $40,000 depreciates 20%
each year.
a) Find the general term, an, of the geometric sequence
that models the value of the sports car at the beginning
of each year.
b) How much is the sports car worth at the beginning of
the fifth year?
a) Find the general term of the geometric sequence that
models the future value of the house.
b) How much is the home worth 5 years after it is purchased? (Hint: Think carefully about what number to
substitute for n.) Round the answer to the nearest dollar.
mes96990_ch15_907-917.qxd
11/9/10
8:26 AM
Page 917
BIA—
Section 15.3 Geometric Sequences and Series
50) A home purchased for $140,000 increases in value by 5%
per year.
4
5
5
3
16 32 64
8, , , , . . .
3 9 27
15 15 15 15
⫺ , ,⫺ , ,...
2 4
8 16
1 1
, , 1, 5, . . .
25 5
45 135
⫺40, ⫺30, ⫺ , ⫺
,...
2
8
72) a1 ⫽ 20, r ⫽ ⫺
a) Find the general term of the geometric sequence that
models the future value of the house.
73) a1 ⫽ 9, r ⫽
74) a1 ⫽ 3, r ⫽
b) How much is the home worth 8 years after it is
purchased? (Hint: Think carefully about what
number to substitute for n.) Round the answer to
the nearest dollar.
75)
76)
Objective 7: Find the Sum of Terms of a Geometric
Sequence
51) Find the sum of the first six terms of the geometric
sequence with a1 ⫽ 9 and r ⫽ 2.
52) Find the sum of the first four terms of the geometric
sequence with a1 ⫽ 6 and r ⫽ 3.
Use the formula for Sn to find the sum of the terms of each
geometric sequence.
VIDEO
71) a1 ⫽ 5, r ⫽ ⫺
53) 7, 28, 112, 448, 1792, 7168, 28672
54) ⫺5, ⫺30, ⫺180, ⫺1080, ⫺6480
1 1
55) ⫺ , ⫺ , ⫺1, ⫺2, ⫺4, ⫺8
4 2
3 3
, , 6, 24, 96, 384
8 2
6 12 24
48
,⫺
58) ⫺3, , ⫺ ,
5 25 125 625
56)
1 1 1 1
57) 1, , , ,
3 9 27 81
7
8
59) a 9(2) i
60) a 5(2) i
5
6
i⫽1
81)
3
2
7 7 7 7
, , , ,...
2 4 8 16
16 32
78) ⫺12, 8, ⫺ , , . . .
3 9
1
80) 36, 6, 1, , . . .
6
82) 4, ⫺12, 36, ⫺108, . . .
Objective 9: Solve an Applied Problem Involving an
Infinite Series
Solve each application.
83) Each time a certain pendulum swings, it travels 75% of
the distance it traveled on the previous swing. If it travels
3 ft on its first swing, find the total distance the pendulum travels before coming to rest.
84) Each time a certain pendulum swings, it travels 70% of
the distance it traveled on the previous swing. If it travels
42 in. on its first swing, find the total distance the pendulum travels before coming to rest.
27 ft
62) a (⫺7)(⫺2) i
i⫽1
6
79)
3
4
85) A ball is dropped from a height of 27 ft. Each time the
2
ball bounces, it rebounds to of its previous height.
3
i⫽1
61) a (⫺4)(3i )
77)
917
i⫽1
i
5
1 i
64) a 2a b
3
1
63) a 3a⫺ b
2
i⫽1
i⫽1
4
i
2
65) a (⫺18)a⫺ b
3
i⫽1
4
2 i
66) a 10a⫺ b
5
i⫽1
67) Gemma decides to save some pennies so that she’ll put 1¢
in her bank on the first day, 2¢ on the second day, 4¢ on
the third day, 8¢ on the fourth day, and so on. If she continued in this way,
a) how many pennies would she have to put in her bank
on the tenth day to continue the pattern?
b) how much money will she have saved after 10 days?
68) The number of bacteria in a culture doubles every day. If
a culture begins with 1000 bacteria, how many bacteria
are present after 7 days?
VIDEO
a) Find the height the ball reaches after the fifth bounce.
b) Find the total vertical distance the ball has traveled
when it comes to rest.
86) A ball is dropped from a height of 16 ft. Each time the
3
ball bounces, it rebounds to of its previous height.
4
Objective 8: Find the Sum of Terms of an Infinite
Geometric Sequence
a) Find the height the ball reaches after the fourth
bounce.
Find the sum of the terms of the infinite geometric sequence,
if possible.
b) Find the total vertical distance the ball has traveled
when it comes to rest.
69) a1 ⫽ 8, r ⫽
1
4
70) a1 ⫽ 18, r ⫽
1
3