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 ·4+ 2 ·4+ 3 ·4+ . . . +4n =4n(n + 1)
2
1)
A statement Sn about the positive integers is given. Write statements Sk and Sk+
1, simplifying Sk+
1 completely.
2)
Sn: 2 is a factor of n2+9n
2)
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.
3)
Sn: 12+42+72+. . . + (3n – 2)2=n(6n2– 3n – 1)
2
3)
4)
Sn: 3+8+13 + . . . + (5n –2) =n(5n + 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)
5+5
6+5
36 + . . . +5
6n – 1 =61 –1
6n
6)
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these
statements is true.
7)
Sn: 2 is a factor of n2+7n
7)
Use mathematical induction to prove that the statement is true for every positive integer n.
8)
1+4+7+ . . . + (3n –2) =n(3n – 1)
2
8)
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.
9)
Sn: 1 ·2 + 2 ·3 + 3 ·4 +. . . + n(n + 1) =n(n + 1)(n + 2)
3
9)
A statement Sn about the positive integers is given. Write statements Sk and Sk+
1, simplifying Sk+
1 completely.
10)
Sn: 1 ·2 + 2 ·3 + 3 ·4 +. . . + n(n + 1) =n(n + 1)(n + 2)
3
11)
Sn: 12+42+72+. . . + (3n – 2)2=n(6n2– 3n – 1)
2
5
Use mathematical induction to prove that the statement is true for every positive integer n.
12)
2+7+12 + . . . + (5n –3) =n(5n – 1)
2
6
13)
10 + 20 + 30 + . . . + 10n =5n(n + 1)
A statement Sn about the positive integers is given. Write statements Sk and Sk+
1, simplifying Sk+
1 completely.
14)
Sn: 1+4+7+ . . . + (3n –2) =n(3n – 1)
2
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the given binomial coefficient.
15)
8
1
15)
A)
1
B)
8
C)
8
7
D)
8!
Use the formula for the sum of the first n terms of a geometric sequence to solve.
16)
Find the sum of the first 9 terms of the geometric sequence: –5, –10, –20, –40, –80, . . . .
16)
A)
–2553
B)
–2518
C)
–2575
D)
–2555
Find the indicated sum.
17)
4
i = 1
2i
17)
A)
20
B)
30
C)
14
D)
18
Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.
18)
a + ar + ar2+ . . . + ar14
18)
A)
15
i = 1
ari – 1
B)
14
i = 1
(ar)i – 1
C)
14
i = 1
ari
D)
14
i = 1
(ar)i
A class is collecting data on eye color and gender. They organize the data they collected into a table. Numbers in the table
represent the number of students from the class that belong to each of the categories. Use the data to solve the problem.
Express probabilities as simplified fractions.
19)
Find the probability that a randomly selected student has brown eyes or green eyes.
Brown Blue Green
Male 22 14 14
Female 20 18 12
19)
A)
21
50
B)
37
50
C)
17
25
D)
13
50
Write the first four terms of the sequence whose general term is given.
20)
an=3n – 3
20)
A)
0, 1, 2, 3
B)
0, 3, 6, 9
C)
0, –3, –6, –9
D)
6, 9, 12, 15
Find the indicated sum.
21)
5
i = 1
(i + 2)!
(i – 1)!
21)
A)
211
B)
25
2
C)
53
4
D)
420
Find the probability.
22)
A bag contains 22 marbles, of which 5 are blue and 10 are green. One marble is drawn from the
bag. What is the probability that the marble drawn is not blue?
22)
A)
5
22
B)
1
3
C)
15
22
D)
17
22
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.
23)
Find a11 when a1= – 2, r =3.
23)
A)
–118,094
B)
–118,098
C)
28
D)
–354,294
Solve the problem.
24)
Ron finds 9 books at a bookstore that he would like to buy, but he can afford only 5 of them. In how
many ways can he make his selection? How many ways can he make his selection if he decides that
one of the books is a must?
24)
A)
15,120; 1680
B)
7560; 840
C)
3024; 1680
D)
126; 70
Provide an appropriate response.
25)
Express the sum using summation notation. Use 1 as the lower limit of summation and i for the
index of summation.
4
5+5
6+6
7+ . . . +23
24
25)
A)
20
i = 1
i
i +3
B)
20
i = 1
i
i +4
C)
20
i = 1
i +3
i +4
D)
21
i = 1
i +3
i +4
Express the sum using summation notation. Use a lower limit of summation, not necessarily 1, and k for the index of
summation.
26)
(a + 1) + (a +b) + (a +b2) + . . . + (a +bn)
26)
A)
n
k = 0
abk
B)
n – 1
k = 0
(a +bk)
C)
n
k = 1
(a +bk)
D)
n
k = 0
(a +bk)
Find the probability.
27)
A spinner has regions numbered 1 through 18. What is the probability that the spinner will stop on
an even number or a multiple of 3?
27)
A)
15
B)
1
C)
2
3
D)
1
3
28)
A card is drawn from a deck of 52 cards. What is the probability that it is a numbered card (2–10) or
a heart?
28)
A)
33
52
B)
53
52
C)
23
52
D)
10
13
Find the indicated sum.
29)
5
i = 1
(–1)i + 1
(i + 1)!
29)
A)
23
60
B)
53
144
C)
–53
144
D)
–23
60
Write the first four terms of the geometric sequence with the given first term, a1, and common ratio, r.
30)
a1= – 6, r = – 10
30)
A)
–6, 60, –360, 2160, . . .
B)
–6, 60, –600, 6000, . . .
C)
–6, 60, 360, 2160, . . .
D)
–6, 60, 600, 6000, . . .
Use the Binomial Theorem to expand the binomial and express the result in simplified form.
31)
(x + 4)3
31)
A)
x3+ 64
B)
3x +12
C)
x3+ 12x2+ 48x + 64
D)
x3+ 4x2+ 16x + 64
Write the first four terms of the geometric sequence with the given first term, a1, and common ratio, r.
32)
a1= – 1
5, r = – 10
32)
A)
–1
5, 2, – 20, 200, . . .
B)
–1
5, 2
5, 2
25 , 8
5, . . .
C)
–1
5, 2
5, 2
25 , 16, . . .
D)
–1
5, 2, 20, 200, . . .
Write the first three terms in the binomial expansion, expressing the result in simplified form.
33)
(x – 5)17
33)
A)
x17 – 85x16 + 3400x15
B)
x17 + 85x16 + 3400x15
C)
x17 + 85x16 – 3400x15
D)
x17 – 85x16 – 3400x15
Evaluate the given binomial coefficient.
34)
8
8
34)
A)
1
B)
2
C)
40,320
D)
0
Use the Binomial Theorem to expand the binomial and express the result in simplified form.
35)
(x2+5y)4
35)
A)
x6+20x5y +30x4y2+20x2y3+5y4
B)
x8+20x6y +150x4y2+500x2y3+625y4
C)
x8+15x6y +150x4y2+375x2y3+625y4
D)
x6+15x5y +150x4y2+375x2y3+625y4
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.
36)
Find a5 when a1=9, r = – 2.
36)
A)
–144
B)
–288
C)
144
D)
16
Write a formula for the general term (the nth term) of the geometric sequence.
37)
2, – 1, 1
2, –1
4, 1
8, . . .
37)
A)
an=2–1
2
n
B)
an=2–1
4
n – 1
C)
an=2–1
2
n + 1
D)
an=2–1
2
n – 1
Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
38)
50
i = 1
–5i
38)
A)
–6247
B)
–6375
C)
–6502
D)
–6370
Write the first five terms of the arithmetic sequence with the given first term, a1, and common difference, d.
39)
a1=10; d =5
39)
A)
10, 15, 20, 25, 30
B)
15, 20, 25, 30, 35
C)
10, 14, 18, 22, 26
D)
0, 10, 15, 20, 25
Find the probability.
40)
A lottery game has balls numbered 1 through 15. What is the probability of selecting an even
numbered ball or a 7?
40)
A)
7
8
B)
8
15
C)
2
5
D)
7
15
Write the first five terms of the arithmetic sequence with the given first term, a1, and common difference, d.
41)
a1=20; d = – 4
41)
A)
–20, –16, –12, –8, –4
B)
24, 19, 14, 9, 4
C)
20, 16, 12, 8, 4
D)
0, 20, 16, 12, 8
Use the formula for the sum of the first n terms of a geometric sequence to solve.
42)
Find the sum of the first five terms of the geometric sequence: 3
2, 3
4, 3
8, . . . .
42)
A)
–1
16
B)
93
32
C)
93
D)
1
8
Write out the first three terms and the last term of the arithmetic sequence.
43)
8
i=1
(2i + 5)
43)
A)
7+ 9 + 11 + . . . + 21
B)
5– 7 + 9 – . . . + 21
C)
5+ 7 + 9 + . . . + 21
D)
7+ 9 + 17 + . . . + 37
Write the first three terms in the binomial expansion, expressing the result in simplified form.
44)
(3x + 2y)7
44)
A)
7x7y + 6x6y2+ 5x5y3
B)
2187x7+ 10,206x6y + 20,412x5y2
C)
2187x7y + 10,206x6y2+ 20,412x5y3
D)
7x7+ 6x6y + 5x5y2
14
Does the problem involve permutations or combinations? Do not solve.
45)
In a student government election, 6 seniors, 3 juniors, and 2sophomores are running for election.
Students elect four at–large senators. In how many ways can this be done?
45)
A)
permutations
B)
combinations
Write a formula for the general term (the nth term) of the sequence. Then use the formula for an to find the twelfth term
of the sequence.
46)
4, 2, 1, 1
2, . . .
46)
A)
an=41
2
n + 1; a12 =1
2048
B)
an=21
2
n – 1; a12 =1
1024
C)
an=41
2
n – 1; a12 =1
512
D)
an=41
2
n; a12 =1
1024
Solve the problem.
47)
A stack of 8 different cards are shuffled and spread out face down. If 5 cards are turned face up,
how many different 5–card combinations are possible?
47)
A)
56
B)
3360
C)
6720
D)
336
Write the first three terms in the binomial expansion, expressing the result in simplified form.
48)
(x2+ 4)9
48)
A)
x18 + 40x16 + 576x14
B)
x18 + 36x16 + 1152x14
C)
x18 + 36x16 + 576x14
D)
x18 + 40x16 + 1152x14
15
Find the indicated sum.
49)
6
i =3
i!
(i – 1)!
49)
A)
10
B)
6
C)
3
D)
18
Find the common ratio for the geometric sequence.
50)
4, 8
3, 16
9, 32
27 , . . .
50)
A)
3
2
B)
2
C)
2
3
D)
1
3
Use the partial sum formula to find the partial sum of the given arithmetic sequence.
51)
Find the sum of the odd integers between 26 and 64.
51)
A)
855
B)
900
C)
810
D)
945
Use the formula for nPr to evaluate the expression.
52)
7P0
52)
A)
5040
B)
0
C)
1
D)
70
Solve the problem.
53)
The matching section of an exam has 5 questions and 9 possible answers. In how many different
ways can a student answer the 5 questions, if none of the answer choices can be repeated?
53)
A)
126
B)
6048
C)
3024
D)
15,120
Use the formula for nCr to evaluate the expression.
54)
11C7
54)
A)
7920
B)
1,663,200
C)
5,702,400
D)
330
55)
6C0
55)
A)
0
B)
720
C)
60
D)
1
Write the first four terms of the sequence whose general term is given.
56)
an=(–1)n + 1
n +9
56)
A)
–1
10 , 1
11 , –1
12 , 1
13
B)
1
10 , –1
11 , 1
12 , –1
13
C)
1
10 , –1
22 , 1
36 , –1
52
D)
–1
11 , 1
12 , –1
13 , 1
14
Solve the problem.
57)
A combination lock has 45 numbers on it. How many different 3–digit lock combinations are
possible if no digit can be repeated?
57)
A)
14,190
B)
1980
C)
85,140
D)
28,380
Provide an appropriate response.
58)
A job pays a salary of $32,000 for the first year. with an annual increase of 5% per year beginning in
the second year. What is the total salary paid over an 9–year period? (Round to the nearest cent.)
58)
A)
$47,278.57
B)
$402,492.56
C)
$352,850.06
D)
$305,571.48
Solve the problem.
59)
A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it
swings only 70% as far as on the previous swing. How far will it swing altogether before coming to
a complete stop?
59)
A)
71 in.
B)
167 in.
C)
83 in.
D)
143 in.
60)
A restaurant offers a choice of 3 salads, 7 main courses, and 4 desserts. How many possible choices
for a meal are there (including single items)?
60)
A)
145 possible meals
B)
159 possible meals
C)
131 possible meals
D)
98 possible meals
Find the common ratio for the geometric sequence.
61)
9, –0.9, 0.09, –0.009, . . .
61)
A)
0.1
B)
–0.1
C)
–1
D)
–0.01
Solve the problem.
62)
Looking ahead to retirement, you sign up for automatic savings in a fixed–income 401K plan that
pays 7% per year compounded annually. You plan to invest $2500 at the end of each year for the
next 20 years. How much will your account have in it at the end of 20 years? (Round to the nearest
dollar.)
62)
A)
$103,787
B)
$102,489
C)
$104,259
D)
$100,946
Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
63)
35
i = 1
(–6i +7)
63)
A)
–3430
B)
–3377.5
C)
–3255
D)
–3535
Solve the problem.
64)
The population of a town is increasing by 300 inhabitants each year. If its current population is
25,713 and this trend continues, what would its population be in 5 years?
64)
A)
257,070 inhabitants
B)
26,913 inhabitants
C)
128,535 inhabitants
D)
27,213 inhabitants
Express the sum using summation notation. Use a lower limit of summation, not necessarily 1, and k for the index of
summation.
65)
2+5
2+3+7
2+ . . . +8
65)
A)
16
k = 1
k
2
B)
16
k = 2
k
2
C)
16
k =4
k
2
D)
8
k =4
k
2
Find the common ratio for the geometric sequence.
66)
4
3, 16
3, 64
3, 256
3, 1024
3, . . .
66)
A)
10
B)
4
C)
4
3
D)
not geometric
Write the first four terms of the sequence whose general term is given.
67)
an=n + 2
2n – 1
67)
A)
3, 4
3, 1, 6
7
B)
3, –4
3, 1, 6
7
C)
–3, –4
3, 1, 6
7
D)
–3, 4
3, 1, 6
7
Does the problem involve permutations or combinations? Do not solve.
68)
A club elects a president, vice–president, and secretary–treasurer. How many sets of officers are
possible if there are 9 members and any member can be elected to each position? No person can
hold more than one office.
68)
A)
permutations
B)
combinations
Solve the problem.
69)
Lisa has 4 skirts, 6 blouses, and 2 jackets. How many 3–piece outfits can she put together assuming
any piece goes with any other?
69)
A)
12 possible outfits
B)
96 possible outfits
C)
24 possible outfits
D)
48 possible outfits
Use the Binomial Theorem to expand the binomial and express the result in simplified form.
70)
(x + 2y)6
70)
A)
x6+ 12x5y + 48x4y2+ 144x3y3+ 192x2y4+ 192xy5+ 64y6
B)
x6+ 12x5y +24x4y2+ 36x3y3+24x2y4+ 12x y5+2y6
C)
x6+ 12x5y +60x4y2+ 160x3y3+240x2y4+ 192xy5+64y6
D)
x6+ 12x5y + 64x4y2+ 40x3y3+ 64x2y4+ 12xy5+ 2y6
20