Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use mathematical induction to prove that the statement is true for every positive integer n.
1)
1+1
2+1
4+ . . . +1
2n – 1 =21 –1
2n
1)
?
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these
statements is true.
2)
Sn: 12+42+72+. . . + (3n – 2)2=n(6n2– 3n – 1)
2
2)
A statement Sn about the positive integers is given. Write statements Sk and Sk+
1, simplifying Sk+
1 completely.
3)
Sn: 1 ·2 + 2 ·3 + 3 ·4 +. . . + n(n + 1) =n(n + 1)(n + 2)
3
3)
4)
Sn: 12+42+72+. . . + (3n – 2)2=n(6n2– 3n – 1)
2
4)
2
Use mathematical induction to prove that the statement is true for every positive integer n.
5)
2 is a factor of n2– n + 2
5)
6)
1 ·3+ 2 ·3+ 3 ·3+ . . . +3n =3n(n + 1)
2
6)
3
7)
4+9+14 + . . . + (5n –1) =n(5n + 3)
2
7)
4
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these
statements is true.
8)
Sn: 4+9+14 + . . . + (5n –1) =n(5n + 3)
2
8)
9)
Sn: 1 ·2 + 2 ·3 + 3 ·4 +. . . + n(n + 1) =n(n + 1)(n + 2)
3
9)
10)
Sn: 2 is a factor of n2+11n
10)
Use mathematical induction to prove that the statement is true for every positive integer n.
11)
10 + 20 + 30 + . . . + 10n =5n(n + 1)
11)
A statement Sn about the positive integers is given. Write statements Sk and Sk+
1, simplifying Sk+
1 completely.
12)
Sn: 2+5+8+ . . . + (3n –1) =n(3n + 1)
2
12)
13)
Sn: 2 is a factor of n2+7n
13)
6
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
14)
As Sunee improves her algebra skills, she takes 0.9 times as long to complete each homework
assignment as she took to complete the preceeding assignment. If it took her 50 minutes to
complete her first assignment, find how long it took her to complete the fifth assignment. Find the
total time she took to complete her first five homework assignments. (Round to the nearest
minute.)
14)
A)
30 min; 172 min
B)
30 min; 205 min
C)
33 min; 172 min
D)
33 min; 205 min
15)
Ms. Patterson proposes to give her daughter Claire an allowance of $0.10 on the first day of her
13–day vacation, $0.20 on the second day, $0.40 on the third day, and so on. Find the allowance
Claire would receive on the last day of her vacation.
15)
A)
$409.60
B)
$4096.10
C)
$819.20
D)
$1.30
Write the first four terms of the sequence whose general term is given.
16)
an=3
5
n
16)
A)
1, 9
25 , 27
125, 81
625
B)
3
5, 3
10 , 3
15 , 3
20
C)
3
5, 9
25 , 27
125, 81
625
D)
1, 3
5, 9
25 , 27
125
Find the probability.
17)
A card is drawn from a well–shuffled deck of 52 cards. What is the probability of getting a red 7?
17)
A)
1
4
B)
1
26
C)
1
52
D)
1
2
Find the sum of the infinite geometric series, if it exists.
18)
3+3
2+3
4+3
8+ . . .
18)
A)
9
2
B)
6
C)
3
2
D)
does not exist
Find the indicated sum.
19)
Find the sum of the first 53 terms of the arithmetic sequence: 2, 4, 6, 8, . . .
19)
A)
2862
B)
108
C)
2915
D)
2867
Write the first four terms of the sequence defined by the recursion formula.
20)
a1= – 2 and an= – 3an–1 for n 2
20)
A)
–2, 8, –20, 56
B)
–2, 6, –18, 54
C)
–2, –6, –18, –54
D)
2, –6, 18, –54
Use the formula for the sum of the first n terms of a geometric sequence to solve.
21)
Find the sum of the first 14 terms of the geometric sequence: –5, –10, –20, –40, –80, . . . .
21)
A)
–81,915
B)
–81,913
C)
–81,878
D)
–81,935
Write out the first three terms and the last term of the arithmetic sequence.
22)
19
i=1
(4i – 5)
22)
A)
–5– 1 + 3 + . . . + 71
B)
–5+ 1 + 3 – . . . + 71
C)
–1+ 11 + 43 + . . . + 299
D)
–1+ 3 + 7 + . . . + 71
Write the first four terms of the sequence whose general term is given.
23)
an=n2
(n + 1)!
23)
A)
1, 2
3, 1
4, 1
15
B)
1
2, 2
3, 3
8, 2
15
C)
1
2, 2
3, 3
4, 4
5
D)
1, 2
3, 1
2, 2
5
Solve the problem. Round to the nearest hundredth of a percent if needed.
24)
During clinical trials of a new drug intended to reduce the risk of heart attack, the following data
indicate the occurrence of adverse reactions among 1200 adult male trial members. What is the
probability that an adult male using the drug will experience nausea?
Adverse Reaction Number
Heartburn 18
Headache 12
Dizziness 10
Urinary problems 6
Nausea 22
Abdominal pain 17
24)
A)
1.42%
B)
25.88%
C)
1.83%
D)
1.71%
Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence
with the given first term, a1, and common difference, d.
25)
Find a23 when a1=24, d = – 3.
25)
A)
–45
B)
90
C)
–42
D)
–66
Write the first five terms of the geometric sequence.
26)
an= – 8an–1; a1=5
26)
A)
–40, 320, –2560, 20,480, –163,840
B)
5, –40, 320, –2560, 20,480
C)
5, –3, –11, –19, –27
D)
–8, –40, –320, –2560, –20,480
Find the indicated sum.
27)
5
i = 3
(i2+ 3)
27)
A)
33
B)
21
C)
70
D)
59
Find the probability.
28)
Two 6–sided dice are rolled. What is the probability the sum of the two numbers on the die will be
5?
28)
A)
5
6
B)
4
C)
1
9
D)
8
9
29)
A card is drawn from a well–shuffled deck of 52 cards. What is the probability of drawing an ace or
a 9?
29)
A)
5
13
B)
2
13
C)
13
2
D)
10
Use the Binomial Theorem to expand the binomial and express the result in simplified form.
30)
(x + 4)3
30)
A)
x3+ 4x2+ 16x + 64
B)
3x +12
C)
x3+ 64
D)
x3+ 12x2+ 48x + 64
10
Evaluate the expression.
31)
5P3
3! –5C3
31)
A)
0
B)
120
C)
6
D)
2
Write a formula for the general term (the nth term) of the arithmetic sequence. Then use the formula for an to find a20,
the 20th term of the sequence.
32)
an=an–1+5, a1=15
32)
A)
an=15n – 10, a20 =290
B)
an=5n +20, a20 =120
C)
an=5n + 10, a20 =110
D)
an=10n +5, a20 =205
Write the first five terms of the geometric sequence.
33)
an=2an–1; a1=4
33)
A)
2, 8, 16, 32, 64
B)
4, 6, 8, 10, 12
C)
8, 16, 32, 64, 128
D)
4, 8, 16, 32, 64
Find the indicated sum.
34)
Find the sum of the odd integers between 202 and 56.
34)
A)
9426
B)
19,092
C)
9417
D)
9431
Find the probability.
35)
A game spinner has regions that are numbered 1 through 9. If the spinner is used twice, what is the
probability that the first number is a 3 and the second is a 5?
35)
A)
8
81
B)
5
8
C)
1
81
D)
1
18
Evaluate the factorial expression.
36)
10!
8!
36)
A)
2!
B)
10
C)
10
8
D)
90
Solve the problem. Round to the nearest dollar if needed.
37)
Kurt deposits $200 each month into an account paying annual interest of 7% compounded monthly.
How much will his account have in it at the end of 5 years?
37)
A)
$14,448
B)
$14,319
C)
$1150
D)
$14,165
Evaluate the factorial expression.
38)
n(n +2 )!
(n +3 )!
38)
A)
1
n +3
B)
n
3
C)
n
(n +3)!
D)
n
n +3
Find the probability.
39)
What is the probability that a card drawn from a deck of 52 cards is not a spade?
39)
A)
3
4
B)
4
13
C)
2
5
D)
1
4
If the given sequence is a geometric sequence, find the common ratio.
40)
4, –12, 36, –108, 324
40)
A)
3
B)
not a geometric sequence
C)
–3
D)
–16
Find the probability.
41)
A card is drawn from a deck of 52 cards. What is the probability that it is a 5 or a club?
41)
A)
2
13
B)
4
13
C)
25
26
D)
17
52
B
D)
Solve the problem.
42)
How many 3–letter codes can be formed using the letters A, B, C, D, and E? No letter can be used
more than once.
42)
A)
10
B)
60
C)
40
D)
20
B
D)
Evaluate the expression.
43)
1 –5P2
8P5
43)
A)
1
336
B)
335
336
C)
23
28
D)
337
336
B
D)
C
D)
Find the probability.
44)
A card is drawn from a well–shuffled deck of 52 cards. What is the probability that the card will
have a value of 4 and be a face card?
44)
A)
0
B)
4
13
C)
1
13
D)
3
13
Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
45)
4
i = 1
4
3
i + 1
45)
A)
12496
243
B)
932
81
C)
700
81
D)
2800
243
Write the first five terms of the arithmetic sequence.
46)
an=an – 1 +3
5; a1= – 1
5
46)
A)
–1
5, –4
5, –7
5, – 2, –13
5
B)
–1
5, –2
5, –3
5, –4
5, – 1
C)
–1
5, 2
5, 6
5, 9
5, 12
5
D)
–1
5, 2
5, 1, 8
5, 11
5
Does the problem involve permutations or combinations? Do not solve.
47)
The matching section of an exam has 4 questions and 10 possible answers. In how many different
ways can a student answer the 4 questions, if none of the answer choices can be repeated?
47)
A)
permutations
B)
combinations
Find the term indicated in the expansion.
48)
(5x + 3)5; 5th term
48)
A)
675x
B)
2025x
C)
1215
D)
3375x2
Find the probability.
49)
A 6–sided die is rolled. What is the probability of rolling a number less than 2?
49)
A)
1
6
B)
1
9
C)
1
3
D)
5
6
Find the indicated sum.
50)
5
i =2
(2i – 2)
50)
A)
18
B)
16
C)
20
D)
12
Find the term indicated in the expansion.
51)
(2x + 3y)11; 8th term
51)
A)
3,849,120x4y8
B)
11,547,360x4y7
C)
5,773,680x7y4
D)
3,849,120x7y4
Write the first four terms of the sequence defined by the recursion formula.
52)
a1=3 and an=3an–1– 3 for n 2
52)
A)
3, 9, 27, 81
B)
3, 12, 39, 120
C)
3, 6, 24, 78
D)
3, 6, 15, 42
Solve the problem.
53)
The matching section of an exam has 3 questions and 7 possible answers. In how many different
ways can a student answer the 3 questions, if none of the answer choices can be repeated?
53)
A)
840
B)
35
C)
1680
D)
210
Write a formula for the general term (the nth term) of the geometric sequence.
54)
–1
5, 1
10 , –1
20 , 1
40 , . . .
54)
A)
an= – 1
2–1
5
n – 1
B)
an= – 1
5–1
2
n – 1
C)
an= – 1
5
n – 1
+3
10
D)
an= – 1
5–1
2(n – 1)
Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.
55)
2+4+6+ . . . +10
55)
A)
5
i = 0
2i
B)
5
i = 1
2i2
C)
5
i = 1
i2
D)
5
i = 1
2i
Write the first four terms of the sequence defined by the recursion formula.
56)
a1= – 1 and an=an–1– 3 for n 2
56)
A)
1, –2, –5, –8
B)
–1, 0, 3, 6
C)
–1, –4, –7, –10
D)
1, 4, 7, 10
Find the indicated sum.
57)
4
i = 1
( –1
2 )i
57)
A)
15
16
B)
–1
16
C)
5
16
D)
–5
16
Find the term indicated in the expansion.
58)
(x2+y3)8; 5th term
58)
A)
420x6y7
B)
70x6y7
C)
70x8y12
D)
420x8y12
Write the first five terms of the geometric sequence.
59)
a1=3; r = – 3
59)
A)
3, 0, –3, –6, –9
B)
3, 9, 27, –81, 243
C)
–3, –9, 27, –81, 243
D)
3, –9, 27, –81, 243
Write out the first three terms and the last term of the arithmetic sequence.
60)
55
i=1
–4i
60)
A)
–4–16 –64 – . . . –880
B)
–4–8–12 – . . . –220
C)
–4+16 –48 + . . . –880
D)
–1 –4–8– . . . –220
Find the probability.
61)
An urn contains balls numbered 1 through 20. A ball is chosen, returned to the urn, and a second
ball is chosen. What is the probability that the first and second balls will be a 6?
61)
A)
3
10
B)
1
400
C)
3
20
D)
1
10
Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of
summation.
62)
7+8+9+10 + . . . +30
62)
A)
23
k = 1
(k – 1)
B)
29
k =6
(k – 1)
C)
31
k =8
(k – 1)
D)
30
k =7
(k – 1)
Evaluate the given binomial coefficient.
63)
198
2
63)
A)
19,503
B)
198!
196!
C)
196
D)
39,006
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence
with the given first term, a1, and common ratio, r.
64)
Find a6 when a1=8, r = – 5.
64)
A)
–25,000
B)
–3125
C)
25,000
D)
125,000
Solve the problem.
65)
In how many ways can Susan arrange 8 books into 3 slots on her bookshelf?
65)
A)
336
B)
1344
C)
6720
D)
13,440
Find the term indicated in the expansion.
66)
(x – 3y)12; 4th term
66)
A)
1980x3y9
B)
–5940x9y3
C)
–5940x3y9
D)
1980x9y4
B
Find the probability.
67)
Given the sequence 5, 6, 7, 8 … 44, what is the probability that a number in the sequence is even or
that it is greater than 8 and less then 25?
67)
A)
55
78
B)
9
10
C)
7
10
D)
11
16
C
68)
What is the probability that a card drawn from a deck of 52 cards is not red?
68)
A)
2
B)
3
4
C)
25
52
D)
1
2
D
If the given sequence is a geometric sequence, find the common ratio.
69)
3, 12, 48, 192, 768
69)
A)
12
B)
4
C)
1
4
D)
not a geometric sequence
B
A
Find the indicated sum.
70)
6
i =3
i!
(i – 1 )!
70)
A)
18
B)
10
C)
6
D)
3
Solve the problem.
71)
A theater has 28 rows with 26 seats in the first row, 30 in the second row, 34 in the third row, and so
forth. How many seats are in the theater?
71)
A)
4592 seats
B)
2240 seats
C)
4480 seats
D)
2296 seats
72)
A student must choose 1 of 4 science electives, 1 of 5 social studies electives, and 1 of 8 language
electives. How many possible course selections are there?
72)
A)
20 course selections
B)
320 course selections
C)
160 course selections
D)
17 course selections
73)
How many 2–letter codes can be formed using the letters A, B, C, D, E, and F? No letter can be used
more than once.
73)
A)
720
B)
30
C)
15
D)
360
Find the indicated sum.
74)
5
i = 1
(i + 2)
74)
A)
25
B)
10
C)
7
D)
18
20