2
14+ 4 = 30
2
30 + 5 = 55
2
Mid-Chapter Quiz: Lessons 10-1 through 10-3
Find the next four terms of each sequence.
55 + 6 = 91
2
91 + 7 = 140.
4. –2187, 729, –243, 81, …
1. 109, 97, 85, 73, …
SOLUTION: SOLUTION: These terms appear to be divided by –3. These terms appear to decrease by 12. Check.
109 – 97 = 12 97 – 85 = 12
85 – 73 = 12
The next four terms are
73 – 12 = 61 61 – 12 = 49
49 – 12 = 37
37 – 12 = 25.
The next four terms are:
–2187 ÷ 729 = –3
729 ÷ –243 = –3
–243 ÷ 81 = –3
81 ÷ (–3) = –27
–27 ÷ (–3) = 9
9 ÷ (–3) = –3
–3 ÷ (–3) = 1.
5. NATURE A petting zoo starts a population of
2. 2, 6, 14, 30, …
SOLUTION: If we subtract each term from the term that follows,
we see a pattern.
6– 2=4
14 – 6 = 8
30 – 14 = 16
n
It appears that each term is generated by adding 2 .
The next four terms are
5
30 + 2 = 62 6
62 + 2 = 126 7
126 + 2 = 254
8
254 + 2 = 510. rabbits with one newborn male and one newborn
female. Assuming that each adult pair will produce
one male and one female offspring per month
starting at two months, how many rabbits will there
be after 6 months?
SOLUTION: During the first two months, there will be one rabbit
couple. At the end of the second month, the pair will
produce a new rabbit couple, making the total for the
third month 2 couples or four rabbits. The new rabbit
couple will grow and develop two months before
producing a new rabbit couple of their own, but the
original pair will now produce a new rabbit couple
each month. The following table shows the pattern.
It is easier to think of the rabbits in terms of couples
and then multiply by 2 at the end.
3. 0, 1, 5, 14, …
Month
Rabbit
Couples
Rabbits
SOLUTION: If we subtract each term from the term that follows,
we see a pattern.
1– 0=1
5– 1=4
14 – 5 = 9
2
It appears that each term is generated by adding n .
The next four terms are:
2
14+ 4 = 30
2
30 + 5 = 55
2
55 + 6 = 91
2
91 + 7 = 140.
eSolutions
Manual
by Cognero
4. –2187,
729,- Powered
–243, 81,
…
SOLUTION: 1
1
2
1
3
2
4
3
5
5
6
8
2
2
4
6
10
16
After 6 months there are 16 rabbits.
Determine whether each sequence is
convergent or divergent.
6. 3, 5, 8, 12, …
SOLUTION: 5– 3=2
8– 5=3
12 – 8 = 4
This sequence is increasing by successively larger
numbers. If we continue this sequence, the nextPage 1
terms get larger and larger. These terms do not
approach a finite number. Therefore, the sequence is
divergent.
Rabbit
Couples
Rabbits
1
1
2
3
5
8
2
2
4
6
10
16
Mid-Chapter
Quiz: Lessons 10-1 through 10-3
After 6 months there are 16 rabbits.
Determine whether each sequence is
convergent or divergent.
6. 3, 5, 8, 12, …
SOLUTION: 5– 3=2
8– 5=3
12 – 8 = 4
This sequence is increasing by successively larger
numbers. If we continue this sequence, the next
terms get larger and larger. These terms do not
approach a finite number. Therefore, the sequence is
divergent.
7. a 1 = 15, a n =
SOLUTION: The first term in this sequence is 15. Find several
more terms using the given recursive formula.
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Page 2
Each term in this sequence is
the previous term.
If we continue this sequence, the next terms would
be 3, 1.5, 0.75, 0.375, 0.1875, 0.09375. These terms
slowly approach a finite number, 0. Therefore, the
sequence is convergent.
Mid-Chapter Quiz: Lessons 10-1 through 10-3
9. a n = n 2 + 5n
SOLUTION: 2
a n = n + 5n
Find several terms in the sequence using the given
explicit formula.
These terms are increasing and do not approach a
finite number. Therefore, the sequence is divergent.
Find each sum.
10. SOLUTION: These terms slowly approach a finite number, –0.5.
Therefore, the sequence is convergent.
8. 48, 24, 12, 6, …
SOLUTION: Each term in this sequence is
the previous term.
11. If we continue this sequence, the next terms would
be 3, 1.5, 0.75, 0.375, 0.1875, 0.09375. These terms
slowly approach a finite number, 0. Therefore, the
sequence is convergent.
SOLUTION: 9. a n = n 2 + 5n
SOLUTION: 2
a n = n + 5n
Find several terms in the sequence using the given
explicit formula.
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12. SOLUTION: Page 3
Mid-Chapter Quiz: Lessons 10-1 through 10-3
14. GOLF In a charity golf tournament, each of the top
12. SOLUTION: ten finishers wins a donation to the charity of his or
her choice. The amount of the donation follows the
arithmetic sequence shown below. What is the total
amount of money donated to charity as a result of
the tournament?
13. SOLUTION: SOLUTION: 14. GOLF In a charity golf tournament, each of the top
ten finishers wins a donation to the charity of his or
her choice. The amount of the donation follows the
arithmetic sequence shown below. What is the total
amount of money donated to charity as a result of
the tournament?
Use the formula for the sum of a finite arithmetic
series.
The total amount of money donated to charity is
$13,750.
Write an explicit formula and a recursive
formula for finding the nth term of each
arithmetic sequence.
15. –11, –15, –19, –23, …
SOLUTION: First, find the common difference.
–15 – (–11) = –4
–19 – (–15) = –4
For an explicit formula, substitute a 1 = –11 and d = –
SOLUTION: 4 in the formula for the nth term of an arithmetic
sequence.
Use the formula for the sum of a finite arithmetic
series.
The total amount of money donated to charity is
$13,750.
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Write an explicit formula and a recursive
formula for finding the nth term of each
arithmetic sequence.
For the recursive formula, state the first term a 1 and
then indicate that the next term is the sum of the
previous term a n – 1 and d.
a n = –11 – 4(n – 1); a 1 = –11, a n = a n – 1 – 4
Page 4
The total amount
of money
donated
to charity
is
Mid-Chapter
Quiz:
Lessons
10-1
through
10-3
$13,750.
Write an explicit formula and a recursive
formula for finding the nth term of each
arithmetic sequence.
15. –11, –15, –19, –23, …
SOLUTION: First, find the common difference.
–15 – (–11) = –4
–19 – (–15) = –4
For an explicit formula, substitute a 1 = –11 and d = –
For the recursive formula, state the first term a 1 and
then indicate that the next term is the sum of the
previous term a n – 1 and d.
a n = –96 + 12(n – 1); a 1 = –96, a n = a n – 1 + 12
17. 7, 10, 13, 16, …
SOLUTION: First, find the common difference.
10 – 7 = 3
13 – 10 = 3
For an explicit formula, substitute a 1 = 7 and d = 3 in
the formula for the nth term of an arithmetic
sequence.
4 in the formula for the nth term of an arithmetic
sequence.
For the recursive formula, state the first term a 1 and
then indicate that the next term is the sum of the
previous term a n – 1 and d.
a n = –11 – 4(n – 1); a 1 = –11, a n = a n – 1 – 4
16. –96, –84, –72, –60, …
For the recursive formula, state the first term a 1 and
then indicate that the next term is the sum of the
previous term a n – 1 and d.
a n = 7 + 3(n – 1); a 1 = 7, a n = a n – 1 + 3
18. 32, 30, 28, 26, …
SOLUTION: First, find the common difference.
30 – 32 = –2
28 – 30 = –2
SOLUTION: First, find the common difference.
–84 – (–96) = 12
–72 – (–84) = 12
For an explicit formula, substitute a 1 = 32 and d = –2
in the formula for the nth term of an arithmetic
sequence.
For an explicit formula, substitute a 1 = –96 and d =
12 in the formula for the nth term of an arithmetic
sequence.
For the recursive formula, state the first term a 1 and
then indicate that the next term is the sum of the
previous term a n – 1 and d.
a n = –96 + 12(n – 1); a 1 = –96, a n = a n – 1 + 12
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7, 10, 13, 16, …
SOLUTION: For the recursive formula, state the first term a 1 and
then indicate that the next term is the sum of the
previous term a n – 1 and d.
a n = 32 – 2(n – 1); a 1 = 32, a n = a n – 1 – 2
19. JEWELRY Mary Anne is hosting a jewelry party.
For each guest who buys an item of jewelry, she
gets a hostess bonus in the amount shown. She
receives a larger amount for each guest making a
purchase.
Page 5
a n = 32 – 2(n – 1); a 1 = 32, a n = a n – 1 – 2
19. JEWELRY Mary Anne is hosting a jewelry party.
For each guestQuiz:
who buys
an item of
jewelry,
she
Mid-Chapter
Lessons
10-1
through
gets a hostess bonus in the amount shown. She
receives a larger amount for each guest making a
purchase.
Mary Anne will reach $100 after 5 guests make
purchases.
10-3
Write an explicit formula and a recursive
formula for finding the nth term of each
geometric sequence.
20. ,
,
,…
,
SOLUTION: First, find the common ratio.
a. How much will Mary Anne receive for the 12th
guest who makes a purchase?
b. If she wants a total hostess bonus of $100, how
many guests need to make a purchase?
SOLUTION: a. Find the 12th term of the arithmetic sequence 10,
15, 20, … The common difference is 5.
÷
= 3
÷
= 3
For an explicit formula, substitute a 1 =
and r = 3
in the nth term formula.
Use the formula for the nth term of an arithmetic
sequence.
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
Mary Anne will receive $65 for the 12th guest who
makes a purchase.
an =
n –1
(3)
; a1 =
, a n = 3a n – 1
b. Use the formula for the sum of a finite arithmetic
series and solve for n.
21. 9, –3, 1,
,…
SOLUTION: First, find the common ratio.
–3 ÷ 9 =
1 ÷ (–3) =
For an explicit formula, substitute a 1 = 9 and r =
in the nth term formula.
Mary Anne will reach $100 after 5 guests make
purchases.
Write an explicit formula and a recursive
formula for finding the nth term of each
geometric sequence.
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20. Manual - Powered by Cognero
,
,
,
SOLUTION: ,…
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
Page 6
an = 9
; a 1 = 9, a n =
an – 1
indicate that the next term is the product of the first
term a n – 1 and r.
n –1
a n = (3)
; a 1 = , a n = 3a n – 1
Mid-Chapter
Quiz:
Lessons 10-1 through 10-3
21. 9, –3, 1,
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
n –1
a n = 3(6)
; a 1 = 3, a n = 6a n – 1
23. –4, 20, –100, 500, …
,…
SOLUTION: SOLUTION: First, find the common ratio.
20 ÷ (–4) = –5
–100 ÷ 20 = –5
First, find the common ratio.
–3 ÷ 9 =
For an explicit formula, substitute a 1 = –4 and r = –5
1 ÷ (–3) =
in the nth term formula.
For an explicit formula, substitute a 1 = 9 and r =
in the nth term formula.
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
n –1
a n = –4(–5)
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
an = 9
; a 1 = 9, a n =
an – 1
22. 3, 18, 108, 648, …
SOLUTION: First, find the common ratio.
18 ÷ 3 = 6
108 ÷ 18 = 6
For an explicit formula, substitute a 1 = 3 and r = 6 in
the nth term formula.
; a 1 = –4, a n = –5a n – 1
24. POPULATION The population of Sandy Shores is
currently 55,000 and is decreasing at a rate of 3%
annually.
a. Write an explicit formula for finding the population
of Sandy Shores during the nth year.
b. What do you predict will be the population of
Sandy Shores after 10 years?
c. After how many years do you predict the
population of Sandy shores will reach 37,000?
SOLUTION: The values of a 1 and r are required to write an
explicit formula for a geometric sequence.
a 1 = 55,000 100% − 3% = 97% retained, so r = 0.97
Substitute the values for a 1 and r in the nth term
formula for a geometric sequence.
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
n –1
a n = 3(6)
; a 1 = 3, a n = 6a n – 1
23. –4, 20, –100, 500, …
an = a1 r
n –1
n –1
a n = 55,000(0.97)
An explicit formula for the population of Sandy
Shores after n years is a n = 55,000(0.97)
n –1
Use the formula found in part a with n = 10.
SOLUTION: First, find the common ratio.
20 ÷ (–4) = –5
–100 ÷ 20 = –5
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For an explicit formula, substitute a 1 = –4 and r = –5
The population of Sandy Shores after 10 years is
Page 7
approximately 41,813.
Use the formula found in part a with a n = 37,000.
For a recursive formula, state the first term a 1. Then
indicate that the next term is the product of the first
term a n – 1 and r.
n –1
Mid-Chapter
10-1 through 10-3
a = –4(–5) Quiz:
; a = Lessons
–4, a = –5a
n
1
n –1
n
24. POPULATION The population of Sandy Shores is
currently 55,000 and is decreasing at a rate of 3%
annually.
a. Write an explicit formula for finding the population
of Sandy Shores during the nth year.
b. What do you predict will be the population of
Sandy Shores after 10 years?
c. After how many years do you predict the
population of Sandy shores will reach 37,000?
The population of Sandy Shores will be 37,000 after
approximately 14 years.
25. MULTIPLE CHOICE If possible, find the sum of
the geometric series 12 + 3 +
+ + . . ..
A 13.5
B 16
C 18
D not possible
SOLUTION: SOLUTION: The values of a 1 and r are required to write an
explicit formula for a geometric sequence.
a 1 = 55,000 Find the common ratio of the related geometric
sequence. 12, 3, , , … .
100% − 3% = 97% retained, so r = 0.97
Substitute the values for a 1 and r in the nth term
formula for a geometric sequence.
an = a1 r
n –1
n –1
a n = 55,000(0.97)
An explicit formula for the population of Sandy
Shores after n years is a n = 55,000(0.97)
Since r < 1, this series has a sum.
Use the formula for the sum of an infinite geometric
series.
n –1
Use the formula found in part a with n = 10.
The correct answer is B.
The population of Sandy Shores after 10 years is
approximately 41,813.
Use the formula found in part a with a n = 37,000.
The population of Sandy Shores will be 37,000 after
approximately 14 years.
25. MULTIPLE CHOICE If possible, find the sum of
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the geometric series 12 + 3 +
A 13.5
+ + . . ..
Page 8