Ch. 13 Sequences; Induction; the Binomial Theorem
13.1 Sequences
1 Write the First Several Terms of a Sequence
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the factorial expression.
1) 5!
A) 120 B) 20 C) 10 D) 9
2) 10!
8!
A) 90 B) 2! C) 10
8D) 10
3) 4!
3!
A) 4 B) 1 C) 4
3D) 4!
4) 5!
7!
A) 1
42 B) 42 C) 2! D) 1
2!
5) 3! 5!
4!
A) 30 B) 1
4C) 5 D) 21
4
6) 7!
5! 2!
A) 21 B) 7 C) 0 D) 1
7) 6!
3! 3!
A) 20 B) 40 C) 120 D) 10
Write out the first five terms of the sequence.
8) {sn} = {n – 6}
A) s1= –5
,
s2 = –4
,
s3= –3
,
s4 = –2
,
s5= –1B)s
1= –6
,
s2= –5
,
s3= –4
,
s4= –3
,
s5= –2
C) s1= 1
,
s2= 2
,
s3= 3
,
s4= 4
,
s5= 5D)s
1=6
,
s2=12
,
s3= 18
,
s4= 24
,
s5=30
9) {sn} = {2n – 3}
A) s1= –1
,
s2= 1
,
s3= 3
,
s4= 5
,
s5= 7B)s
1= –1
,
s2=0
,
s3= 1
,
s4= 2
,
s5= 3
C) s1= 5
,
s2= 7
,
s3= 9
,
s4= 11
,
s5= 13 D) s1=1
,
s2= –1
,
s3= –3
,
s4= –5
,
s5= –7
Page 1
10) {sn} = {3(3n – 2)}
A) s1= 3
,
s2= 12
,
s3= 21
,
s4= 30
,
s5=39 B) s1=1
,
s2=4
,
s3=7
,
s4= 10
,
s5= 13
C) s1= –6
,
s2= 3
,
s3= 12
,
s4= 21
,
s5=30 D) s1=3
,
s2=6
,
s3=9
,
s4= 12
,
s5= 15
11) {sn} = {n2 – n}
A) s1= 0, s2= 2, s3= 6, s4= 12, s5= 20 B) s1=2, s2=6, s3=12, s4= 20, s5=30
C) s1= 1, s2= 4, s3= 9, s4= 16, s5= 25 D) s1=0, s2=3, s3=8, s4= 15, s5= 24
12) {sn} = {2n}
A) s1= 2
,
s2= 4
,
s3= 8
,
s4= 16
,
s5= 32 B) s1=1
,
s2=4
,
s3=9
,
s4= 16
,
s5= 25
C) s1= 1
,
s2= 2
,
s3= 4
,
s4= 8
,
s5= 16 D) s1=4
,
s2=8
,
s3=16
,
s4= 32
,
s5=64
13) {cn} = 2n + 1
2n
A) c1 = 3
2, c2 = 5
4, c3 = 7
6, c4 = 9
8, c5 = 11
10 B) c1 = 5
4, c2 = 7
6, c3 = 9
8, c4 = 11
10, c5 = 13
12
C) c1 = 3
2, c2 = 5
2, c3 = 7
5, c4 = 9
2, c5 = 11
2D) c1 = 5
2, c2 = 7
2, c3 = 9
2, c4 = 11
2, c5 = 13
2
14) {sn} = 2n
4n + 2
A) s1 = 1
3, s2 = 2
9, s3 = 4
33, s4 = 8
129 , s5 = 16
513 B) s1 = 1
3, s2 = 2
5, s3 = 3
7, s4 = 4
9, s5 = 5
11
C) s1 = 2
9, s2 = 4
33, s3 = 8
129 , s4 = 16
513 , s5 = 32
2049 D) s1 = 2
5, s2 = 3
7, s3 = 4
9, s4 = 5
11, s5 = 6
13
15) {cn} = (–1)n
(n + 4)(n + 5)
A) c1 = – 1
30, c2 = 1
42, c3 = – 1
56, c4 = 1
72, c5 = – 1
90
B) c1 = 1
30, c2 = – 1
42, c3 = 1
56, c4 = – 1
72, c5 = 1
90
C) c1 = – 1
11, c2 = 1
13, c3 = – 1
15, c4 = 1
17, c5 = – 1
19
D) c1 = 1
11, c2 = – 1
13, c3 = 1
15, c4 = – 1
17, c5 = 1
19
16) {cn} = 5n
n
A) c1 = 5, c2 = 25
2, c3 = 125
3, c4 = 625
4, c5 = 625 B) c1 = 0, c2 = 5, c3 = 25
2, c4 = 125
3, c5 = 625
4
C) c1 = 5
,
c2 = 25
,
c3 = 125
,
c4 = 625
,
c5=3125 D) c1=1, c2=5
,
c3= 25
,
c4 = 125
,
c5=625
Page 2
17) {sn} = (
–1)n – 1 n + 3
2n – 1
A) s1= 4, s2= – 5
3, s3= 6
5, s4= – 1, s5= 8
9B) s1= 4, s2= 5
3, s3= 6
5, s4= 1, s5= 8
9
C) s1= –4, s2= 5
3, s3= – 6
5, s4= 1, s5= – 8
9D) s1= –4, s2= 5
3, s3= 6
5, s4= – 1, s5= 8
9
18) {sn} = n
n2 + 2
A) s1= 1
3, s2= 1
3, s3= 3
11, s4= 2
9, s5= 5
27 B) s1= 1
3, s2= 1
3, s3= 3
8, s4= 2
5, s5= 5
12
C) s1= 1
2, s2= 1
3, s3= 3
8, s4= 2
5, s5= 5
12 D) s1= 1
4, s2= 1
3, s3= 3
8, s4= 2
5, s5= 5
12
The given pattern continues. Write down the nth term of the sequence {an} suggested by the pattern.
19) 4
,
12
,
20
,
28
,
36
,
…
A) an = 2(4n – 2 ) B) an = 8n –2C)a
n = 4(8)
n–1D) an = 4n –8
20) 2
,
6
,
10
,
14
,
18
,
…
A) an = 4n – 2B)a
n = 2n –4C)a
n = 2(4)
n–1D) an = n +4
21) 0, 2, 6, 12, 20, …
A) an = n2 – nB)a
n = 2n – 2C)a
n = 4n – 6D)a
n = 2n–1 – 1
22) 2
,
4
,
8
,
16
,
32
,
…
A) an = 2nB) an = 2 n C) an = 2 + 2(n – 1) D) an = 2n–1 + 1
23) 3
,
–6
,
9
,
–12
,
…
A) an = (–1)n + 1 · 3n B) an = (–1)n · 3n C) an = (–1)n +1 · 3D)a
n = (–1)n · 3
24) 1
1 · 3 , 1
2 · 4 , 1
3 · 5 , 1
4 · 6 , …
A) an = 1
n(n + 2) B) an = 1
n · 2n C) an = 1
2n D) n(n +2)
25) 1, 1
4, 1
16, 1
64, …
A) an = 1
4n – 1 B) an = 1
4nC) an = 1
4D) an = 1
4 + n
26) 1
1, 1
4, 1
9, 1
16, 1
25, …
A) an = 1
n2B) an = 1
nn–1C) an = 1
3n – 2 D) an = 1
2
n–1
Page 3
Solve.
27) The number of students in a school in year n is estimated by the model an = 5n2 + 14n + 80. About how
many students are in the school in each of the first three years?
A) 99
,
128
,
167 B) 113
,
142
,
181 C) 104
,
128
,
167 D) 99
,
128
,
152
28) During a five–year period, a company doubles its profits each year. If the profits at the end of the fifth year
are $240,000, then what are the profits for each of the first four years?
A) $15,000
,
$30,000
,
$60,000
,
$120,000 B) $15,000
,
$30,000
,
$60,000
,
$150,000
C) $16,000
,
$32,000
,
$64,000
,
$126,000 D) $15,000
,
$30,000
,
$45,000
,
$60,000
2 Write the Terms of a Sequence Defined by a Recursive Formula
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The sequence is defined recursively. Write the first four terms.
1) a1 = 1; an = an–1 – 4
A) a1 = 1
,
a2 = –3
,
a3 = –7
,
a4 = –11 B) a1= –4
,
a2= –8
,
a3 = –12
,
a4 = –16
C) a1 = 1
,
a2 = 5 , a3 = 9 , a4 = 13 D) a1=1
,
a2= –1, a
3 = –5 , a4 = –9
2) a1 = 4; an = 3an–1
A) a1 = 4
,
a2 = 12
,
a3= 36
,
a4 = 108 B) a1=4
,
a1=11
,
a2 = 10
,
a4 = 9
C) a1 = 9
,
a1 = 27
,
a2= 81
,
a4 = 162 D) a1=4
,
a1=14
,
a2 = 38
,
a4 = 110
3) a1 = 3; an = 3an–1 + 3
A) a1 = 3
,
a2 = 12
,
a3= 39
,
a4 = 120 B) a1=3
,
a2=12
,
a3 = 30
,
a4 = 84
C) a1 = 3
,
a2 = 9
,
a3 = 27
,
a4 = 81 D) a1=3
,
a2=6
,
a3 = 15
,
a4 = 42
4) a1 = 2, a2 = 5; an = an–2 – 3an–1
A) a1 = 2, a2 = 5, a3 = –13, a4 = 44 B) a1=2, a2=5, a3= –1, a4 = –16
C) a1 = 2, a2 = 5, a3 = 1, a4 = 2D)a
1=2, a2=5, a3= 17, a4 = –46
5) a1 = 135; an+1 = 1
3 (an)
A) a1 = 135, a2 = 45, a3 = 15, a4 = 5B)a
1=135, a2=67.5, a3 = 33.75, a4=16.875
C) a1 = 135, a2 = 405, a3 = 1215, a4=3645 D) a1=135, a2=105, a3 = 75, a4 =45
6) a1 = –5; an = n – an – 1
A) a1 = –5
,
a2 = 7
,
a3 = –4
,
a4 = 8B)a
1= –5
,
a2= –3
,
a3 = 6
,
a4 = –2
C) a1 = –5
,
a2 = –10
,
a3 = 5
,
a4 = –15 D) a1= –6
,
a2= –3
,
a3 = –5
,
a4 = –2
7) a1 = –8; an = an – 1
n + 1
A) a1 = –8, a2 = – 8
3, a3 = – 2
3, a4 = – 2
15 B) a1 = –8, a2 = – 4, a3 = – 4
3, a4 = – 1
3
C) a1 = –8, a2 = –8, a3 = – 8
3, a4 = – 2
3D) a1 = –8, a2 = –8, a3 = – 4, a4 = – 4
3
Page 4
8) a1 = y; an = an–1 + U
A) a1 = y
,
a2 = y + U
,
a3 = y + 2U
,
a4=y+3U B) a1=y
,
a2=U
,
a3 = 2U
,
a4 = 3U
C) a1 = y
,
a2 = y – U
,
a3 = y – 2U
,
a4=y–3U D) a1=U
,
a2=U+y
,
a3 = U + 2y
,
a4=U+3y
9) a1 = 10; an = 10an–1
A) a1 = 10, a2 = 10 10, a3 = 10 10 10, a4 = 10 10 10 10
B) a1 = 10, a2 = 10, a3 = 10 10, a4 = 100
C) a1 = 10, a2 = 10 10, a3 = 100 10, a4 = 1000 10
D) a1 = 10, a2 = 10, a3 = 10, a4 = 10
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
10) A wildlife refuge currently has 100 deer in it. A local wildlife society decides to add an additional 2 deer
each month. It is already known that the deer population is growing 12% per year. The size of the
population is given by the recursively defined sequence
p0 = 100 pn = 1.01pn–1 + 2
How many deer are in the wildlife refuge at the end of the second month? That is, what is p2?
11) Lexington Reservoir has 300 million gallons of water. About 0.07% of the water is lost to evaporation every
week. About 150,000 gallons of water enter the reservoir every week. The amount of water in the reservoir
at the end of each week is given by the recursively defined sequence
w0 = 300 wn = (0.9993)wn–1 + 0.15
Determine the amount of water in the reservoir at the beginning of the second week. That is, determine
w2.
3 Use Summation Notation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write out the sum.
1)
n
k = 1
(k + 6)
∑
A) 7 + 8 + 9 + … + (n + 6) B) n +6
C) 6 + 7 + 8 + … + (n + 6) D) 1 +2 +3 +… +n
2)
n
k = 1
(3k + 10)
∑
A) 13 + 16 + 19 + … + (3n + 10) B) 3n +10
C) 10 +13 + 16 + … + (3n + 10) D) 1 +2 +3 +… +n
Page 5
3)
n
k = 1
(k + 3)2
∑
A) 16 + 25 + 36 + … + (n + 3)2B) (n + 3)2
C) 4 + 5 + 6 + … + (n + 3)2D) 1 +2 +3 +… +n
4)
n – 1
k = 0
(2k + 7)
∑
A) 7 + 9 + 11 + … + (2n + 5) B) 9 +11 +13 +… +(2n + 5)
C) 7 + 9 + 11 + … + (2n + 7) D) 9 +11 +13+… +(2n + 7)
5)
n
k = 0
5
9
k
∑
A) 1 + 5
9 + 25
81 + … + 5
9
nB) 5
9 + 25
81 + 125
729 + … + 5
9
n
C) 0 + 5
9 + 25
81 + … + 5
9
nD) 0 +1 +2 +3 +… + n
6)
n – 1
k = 0
1
4k + 1
∑
A) 1
4 + 1
16 + 1
64 + … + 1
4nB) 1 + 1
4 + 1
16 + … + 1
4n
C) 0 + 1
4 + 1
16 + … + 1
4nD) 1
4 + 1
16 + 1
64 + … + 1
4n + 1
7)
n
k = 1
k2
4
∑
A) 1
4 + 1 + 9
4 + … + n2
4B) n2
4
C) 4 + 16 + 36 + … + n2
4D) 1 +2 +3 +… +n
8)
n
k = 1
6k+1
∑
A) 62 + 63 + 64+ … + 6n+1B) 6 + 62 + 63+ … + 6n
C) 62 + 63 + 64+ … + 6nD) 6 + 62 + 63+ … + 6n+1
Express the sum using summation notation.
9) 3 + 6 + 9 + … + 15
A)
5
k = 1
3k
∑
B)
5
k = 0
3k
∑
C)
5
k = 1
k2
∑
D)
5
k = 1
3k2
∑
Page 6
10) 23 + 33 + 43 + … + 73
A)
7
k = 2
k3
∑
B)
7
k = 1
k3
∑
C)
n
k = 2
k3
∑
D)
7
k = 3
( k – 1)3
∑
11) 3 + 12 + 27 + . . . + 75
A)
5
k =1
3k2
∑B)
5
k = 0
3k2
∑C)
5
k =1
k2
∑D)
5
k = 1
32k
∑
12) 1
4 + 2
5 + 1
2 + … + 5
6
A)
15
k = 1
k
k + 3
∑
B)
15
k = 0
k
k + 3
∑
C)
n
k = 1
k
k + 3
∑
D)
15
k = 3
k
k + 1
∑
13) 7 + 72
2 + 73
3 + … + 7n
n
A)
n
k = 1
7k
k
∑B)
n
k = 0
7k
k
∑C)
n
k = 1
7k
k
∑D)
n
k = 0
7k
k
∑
14) 8
e + 16
e2 + 24
e3… + 8n
en
A)
n
k = 1
8k
ek
∑B)
n
k = 0
8k
ek
∑C)
n
k = 1
8 + k
ek
∑D)
n
k = 0
8 + k
ek
∑
15) 4
5 – 16
25 + 64
125 – … + (–1)(11 + 1) 4
5
11
A)
11
k = 1
(–1)(k+1)
∑4
5
k
B)
12
k = 1
(–1)(k+1)
∑4
5
k
C)
n
k = 1
(–1)(k+1)
∑4
5
k
D)
11
k = 1
4
5
k
∑
16) 212
75 + 211
76 + 210
77 + … + 25
712
A)
8
k = 1
213 – k
74 + k
∑B)
4
k = 1
213 +k
74 + k
∑C)
8
k = 1
213 –k
7k
∑D)
8
k = 1
2–k
74 + k
∑
Page 7
17) 42 + 83 + 124 + … + 329
A)
8
k = 1
(4k)k + 1
∑B)
8
k = 1
(4k)k
∑
C)
8
k = 1
2(k – 1)k + 1
∑D)
8
k = 1
4k2k – 1
∑
18) 1
w + s
2w + s2
3w + … + sn–1
nw
A)
n
k=1
sk–1
kw
∑B)
n
k=1
sk
kw
∑C)
n
k=0
sk–1
kw
∑D)
n
k=0
sk
kw
∑
4 Find the Sum of a Sequence Algebraically and Using a Graphing Utility
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the sum of the sequence.
1)
10
k = 1
4
∑
A) 40 B) 4 C) 10 D) 6
2)
7
k = 1
(–k)
∑
A) –28 B) –7C)
–6D)
–42
3)
5
k = 2
6k
∑
A) 84 B) 30 C) 42 D) 54
4)
5
k = 1
(k – 6)
∑
A) –15 B) –1C)
–6D)
–14
5)
5
k = 2
(3k – 4)
∑
A) 26 B) 24 C) 22 D) 17
Page 8
6)
16
k = 1
(2k + 7)
∑
A) 384 B) 53 C) 272 D) 400
7)
5
k = 3
(k2 – 3)
∑
A) 41 B) 40 C) 15 D) 3
8)
4
k = 2
k(k – 12)
∑
A) –79 B) –90 C) –52 D) –18
5 Solve Annuity and Amortization Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Karen has a balance of $1000 on a department store credit card that charges 1.5% interest per month on
any unpaid balance. She can afford to pay $150 toward the balance each month. Her balance each month,
after making a $150 payment, is given by the recursively defined sequence
B0 = $1000 Bn = 1.015Bn–1 – 150
Determine Karen’s balance after making the first payment. That is, determine B1.
A) $865.00 B) $727.98 C) $915.00 D) $965.00
2)
J
ake bought a truck by taking out a loan for $32,600 at 0.25% interest per month. Jake’s regular monthly
payment is $567, but he decides to pay an extra $75 toward the balance each month. His balance each
month, after making his payment, is given by the recursively defined sequence
B0 = $32,600 Bn = 1.0025Bn–1– 642
Determine Jake’s balance after making the first payment. That is, determine B1.
A) $32,039.50 B) $32,681.50 C) $32,114.50 D) $31,964.50
3) Maria deposited $2500 in an independent retirement account at her bank. She earns 0.5% interest per
month on the balance. Each month, she deposits $100 in the account. Her balance each month after making
a $100 deposit, is given by the recursively defined sequence
B0 = $2500 Bn = 1.005Bn–1 + 100
If she made the initial deposit on September 30, and makes each monthly deposit on the last day of the
month, how much money will be in the account at the end of the year for Maria to count as a deduction
for that year’s federal income taxes? That is, determine B3.
A) $2839.19 B) $2612.50 C) $2725.56 D) $3209.31
Page 9
4)
J
ack decided to put $500 into an IRA account every 3 months at a rate of 6% compounded quarterly. Find
a recursive formula that represents his balance at the end of each quarter. How many years will it be
before the value of the account is $100,000? What will be the balance in 30 years when Jack retires?
A) B0 = 500, Bn = 1
+ 0.06
4B(n – 1) + 500; 24 years; $168,129
B) B0 = 500, Bn = 1
+ 0.06
3B(n – 1) + 500; 27 years; $126,050
C) B0 = 500, Bn = 1
+ 0.6
4B(n – 1) + 500; 24 years; $168,129
D) B0 = 500, Bn = 1
+ 0.6
3B(n – 1) + 500; 27 years; $126,050
5) After working for 25 years you would like to have $600,000 in an annuity for early retirement. If the
annual interest rate is 7.5%, compounded monthly, what will your monthly deposit need to be?
A) $683.95 B) $2080.03 C) $839.58 D) $1373.90
6) To save for retirement, you decide to deposit $1500 into an IRA at the end of each year for the next 40
years. If the interest rate is 7% per year compounded annually, find the value of the IRA after 40 years.
Round to the nearest dollar.
A) $299,453 B) $20,962 C) $278,460 D) $4,618,809
7) Looking ahead to retirement, you sign up for automatic savings in a fixed–income 401K plan that pays 5%
per year compounded annually. You plan to invest $2000 at the end of each year for the next 30 years.
How much will your account have in it at the end of 30 years? Round to the nearest dollar.
A) $132,878 B) $134,648 C) $131,335 D) $134,176
8) 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 11 years? Round to the nearest dollar.
A) $39,598 B) $3157 C) $39,444 D) $39,727
9) Domenica invests $200 each quarter in a fixed–interest mutual fund paying annual interest of 7%
compounded quarterly. How much will her account have in it at the end of 6 years? Round to the nearest
dollar.
A) $5902 B) $1431 C) $17,678 D) $6031
10) You deposit $150 every 6 months into an annuity with an annual interest rate of 8%
,
compounded
semiannually. What is the balance after 20 years? What is the balance after 40 years?
A) $14,253.83; $82,686.75 B) $4466.71; $82,686.75
C) $4466.71; $14,253.83 D) $14,253.83; $38,858.48
11) Christine contributes $250 each month to her 401(k). To the nearest dollar, what will be the value of
Christine’s 401(k) in 20 years if the per annum rate of return is assumed to be 10% compounded monthly.
Round to the nearest dollar.
A) $189,842 B) $66,390 C) $62,096 D) $178,851
Page 10
13.2 Arithmetic Sequences
1 Determine if a Sequence is Arithmetic
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
An arithmetic sequence is given. Find the common difference and write out the first four terms.
1) {sn} = {n – 7}
A) d = 1; s1 = –6
,
s2 = –5
,
s3 = –4
,
s4= –3B)d
= –1; s1= –6
,
s2 = –5
,
s3 = –4
,
s4= –3
C) d = 1; s1 = 6
,
s2 =5
,
s3 = 4
,
s4 =3D)d
= –1; s1=6
,
s2 = 5
,
s3 = 4
,
s4=3
2) {sn} = {8 – 4n}
A) d = –4; s1 = 4
,
s2 = 0
,
s3 = –4
,
s4= –8B)d
= –4; s1= –4
,
s2 = –8
,
s3 = –12
,
s4= –16
C) d = 4; s1 = 4
,
s2 =8 , s3 = 12 , s4=16 D) d = –4; s1=4
,
s2 = 2 , s3 = –2 , s4= –6
3) {sn} = {2n + 9}
A) d = 2; s1 = 11
,
s2 = 13
,
s3 = 15
,
s4=17 B) d =2; s1=2
,
s2= 11
,
s3 = 13
,
s4=15
C) d = 9; s1 = 11
,
s2 = 13
,
s3 = 15
,
s4=17 D) d =9; s1=2
,
s2= 11
,
s3 = 13
,
s4=15
4) {sn} = 1
9 + n
8
A) d = 1
8; s1 = 17
72, s2 = 13
36, s3 = 35
72, s4 = 11
18 B) d = 1
9; s1 = 1
9, s2 = 17
72, s3 = 13
36, s4 = 35
72
C) d = 1
8; s1 = 1
9, s2 = 17
72, s3 = 13
36, s4 = 35
72 D) d = 1
9; s1 = 17
72, s2 = 13
36, s3= 35
72, s4 = 11
18
5) {sn} = {ln 7n}
A) d = ln 7; s1 = ln 7
,
s2 = 2 ln 7
,
s3=3 ln 7
,
s4=4 ln 7
B) d = n ln 7; s1 = ln 7
,
s2 = 2 ln 7
,
s3=3 ln 7
,
s4=4 ln 7
C) d = ln 7; s1 = ln 7
,
s2 = ln 14
,
s3 =ln 21
,
s4=ln 28
D) d = n ln 7; s1 = ln 7
,
s2 = ln 14
,
s3=ln 21
,
s4=ln 28
2 Find a Formula for an Arithmetic Sequence
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the nth term and the indicated term of the arithmetic sequence {an} whose initial term, a, and common
difference, d, are given.
1) a1 = 7; d = 6
an = ?; a14 = ?
A) an = 1 + 6n; a14 = 85 B) an=7+6n; a14 = 85
C) an = 1 + 6n; a14 = 61 D) an=1–6n; a14 = 85
2) a1 = 81; d = –4
an = ?; a10 = ?
A) an = 85 – 4n; a10 = 45 B) an=81 –4n; a10 = 45
C) an = 85 – 4n; a10 = 33 D) an=81 –4n; a10 = 33
Page 11
3) a1 = 4; d = –6
an = ?; a15 = ?
A) an = 10 – 6n; a15 = –80 B) an=4–6n; a15 = –80
C) an = 10 – 6n; a15 = –32 D) an=10 +6n; a15 = –80
4) a1 = –8; d = 8
an = ?; a10 = ?
A) an = –16 + 8n; a10 = 64 B) an= –8+8n; a10 = 64
C) an = –16 + 8n; a10 = 112 D) an= –16 –8n; a10 = 64
5) a1 = 0; d = 1
4
an = ?; a28 = ?
A) an = 1
4(n – 1); a28 = 27
4B) an = 1
4n; a28 = 7
C) an = 1
4(n + 1); a28 = 29
4D) an=4(n –1); a28 = 27
6) a1 = 11; d = 11
an = ?; a16 = ?
A) an = 11n; a16 = 16 11 B) an = 11n – 11; a16 = 15 11
C) an = – 11n; a16 = – 16 11 D) an = 11n + 11; a16 = 17 11
Find the indicated term of the arithmetic sequence.
7) The eighteent
h
term of the arithmetic sequence –10
,
–6
,
–2
,
. . .
A) 58 B) 62 C) –78 D) –82
8) The fourteenth term of the arithmetic sequence 0, 15
,
30
,
. . .
A) 195 B) 210 C) 225 D) 182
9) The twenty–second term of the arithmetic sequence 30
,
26
,
22
,
. . .
A) –54 B) –58 C) –84 D) 114
10) The eighteenth term of the arithmetic sequence –14 3, –12 3, –10 3, . . .
A) 20 3 B) 22 3 C) –48 3 D) –50 3
Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
11) 9th term is 78; 13th term is 70
A) a1 = 94
,
d = –2
,
an = an–1 – 2B)a
1=94
,
d =2
,
an = an–1 + 2
C) a1 = 96
,
d = –2
,
an = an–1 – 2D)a
1=96
,
d =2
,
an = an–1 + 2
12) 8th term is –44; 15th term is –93
A) a1 = 5
,
d = –7
,
an = an–1 – 7B)a
1=5
,
d =7
,
an = an–1 + 7
C) a1 = 12
,
d = –7
,
an = an–1 – 7D)a
1=12
,
d =7
,
an = an–1 + 7
Page 12
13) 8th term is 58; 14th term is 112
A) a1 = –5
,
d = 9
,
an = an–1 + 9B)a
1= –5
,
d =9
,
an = an–1 – 9
C) a1 = –14
,
d = 9
,
an = an–1 + 9D)a
1= –14
,
d =9
,
an = an–1 – 9
14) 6th term is –10; 15th term is –46
A) a1 = 10
,
d = –4, an = an–1 – 4B)a
1= –30
,
d =4, an = an–1 + 4
C) a1 = –10
,
d = –4, an = an–1 – 4D)a
1=10
,
d =4, an = an–1 + 4
Solve.
15) Find x so that x – 7
,
2x – 11
,
and 5x – 27 are consecutive terms of an arithmetic sequence.
A) x = 6B)x = –5C)x
= –1D)x
= –13
16) The population of a town is increasing by 300 inhabitants each year. If the town’s starting population was
27,957, what was the population after 10 years?
A) 30,957 inhabitants B) 279,435 inhabitants
C) 30,657 inhabitants D) 558,870 inhabitants
3 Find the Sum of an Arithmetic Sequence
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the sum.
1) (–6) + (–1) + 4 + 9 + … + 39
A) 165 B) 170 C) 160 D) 330
2) 1 + 2 + 3 + … + 863
A) 372816 B) 371953 C) 745,632 D) 744,769
3) 2 + 4 + 6 + … + 1954
A) 955,506 B) 954,529 C) 953,552 D) 956,484
4) 1 + 3 + 5 + … + 1575
A) 620,944 B) 621,732 C) 622,521 D) 619,369
5) 5 + 10 + 15 + … + 290
A) 8555 B) 8410 C) 17405
2D) 8265
6) –3 + 1 + 5 + 9 + 13 + … + (4n – 7)
A) n(2n – 5) B) n(4n –7) C) n(2n +5) D) n(4n +7)
7)
49
n = 1
(2n + 7)
∑
A) 2793 B) 2744 C) 2989 D) 3160.5
8)
39
n = 1
(2n – 5)
∑
A) 1365 B) 1326 C) 1482 D) 1618.5
Page 13
9)
25
n = 1
(–4n + 1)
∑
A) –1275 B) –1225 C) –1150 D) –1062.5
10)
30
n = 1
(–5n – 4)
∑
A) –2445 B) –2370 C) –2325 D) –2220
11)
10
n = 1
(3.4n + 8.75)
∑
A) 274.5 B) 42.75 C) 231.75 D) 192.4
Solve.
12) A theater has 28 rows with 23 seats in the first row, 26 in the second row, 29 in the third row, and so forth.
How many seats are in the theater?
A) 1778 seats B) 1820 seats C) 3556 seats D) 3640 seats
13) A brick staircase has a total of 20 steps The bottom step requires 115 bricks. Each successive step requires 4
less bricks than the prior one. How many bricks are required to build the staircase?
A) 1540 bricks B) 3060 bricks C) 3080 bricks D) 1500 bricks
14) Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of
$1500 per year. How many years will it be before you’ve made a total (or aggregate) salary of $1,025,000?
A) 20 years B) 21 years C) 25 years D) 18 years
15) A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3
additional seats. How many seats are in the civic theater?
A) 1092 seats B) 790 seats C) 1070 seats D) 1010 seats
13.3 Geometric Sequences; Geometric Series
1 Determine if a Sequence is Geometric
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
A geometric sequence is given. Find the common ratio and write out the first four terms.
1) {sn} = 6 1
2
n–1
A) an = 6 1
2
n–1
r = 1
2; s1 = 6, s2 = 3, s3 = 3
2, s4 = 3
4
B) an = 6 · (2)n–1
r = 2; s1 = 6, s2 = 12, s3 = 24, s4 = 48
C) an = 6 1
4
n–1
r = 1
4; s1 = 6, s2 = 3
2, s3 = 3
8, s4 = 3
32
D) an = 1
2 (6)n–1
r = 1
2; s1 = 6, s2 = 12, s3 = 24, s4 = 48
Page 14
2) {sn} = {2n}
A) r = 2; s1 = 2
,
s2 = 4
,
s3 = 8
,
s4 = 16 B) r =2; s1=2
,
s2= 4
,
s3 = 6
,
s4 = 8
C) r = 2n; s1 = 2
,
s2 = 4
,
s3 = 8
,
s4 =16 D) r =2n; s1=2
,
s2= 4
,
s3 = 6
,
s4 =8
3) {tn} = 5
2
n
A) r = 5
2; t1 = 5
2, t2 = 25
4, t3 = 125
8, t4 = 625
16 B) r = 5
2n; t1 = 5
2, t2 = 25
4, t3 = 125
8, t4 = 625
16
C) r = 5
2; t1 = 5
2, t2 = 5
2, t3 = 5
2, t4 = 5
2D) r = 5
2n; t1 = 5
2, t2 = 5
2, t3 = 5
2, t4 = 5
2
4) {dn} = 5n
30
A) r = 5; d1 = 1
6, d2 = 5
6, d3 = 25
6, d4 = 125
6B) r = 5
6; d1 = 1
6, d2 = 5
6, d3 = 25
6, d4 = 125
6
C) r = 5; d1 = 5
6, d2 = 25
6, d3 = 125
6, d4 = 625
6D) r = 5
6; d1 = 5
6, d2 = 25
6, d3 = 125
6, d4 = 625
6
5) {sn} = {33n}
A) r = 27; s1 = 27
,
s2 = 729
,
s3 = 19,683
,
s4=531,441
B) r = 3; s1 = 27
,
s2 =729
,
s3 = 19,683
,
s4=531,441
C) r = 9; s1 = 9
,
s2 = 18
,
s3 = 27
,
s4 =36
D) r = 27; s1 = 9
,
s2 =18
,
s3 = 27
,
s4=36
6) {un} = 4n
5n – 1
A) r = 4
5; u1 = 4, u2 = 16
5, u3 = 64
25, u4 = 256
125 B) r = 4
5; u1 = 4
5, u2 = 16
25, u3 = 64
125 , u4 = 256
625
C) r = 4; u1 = 4, u2 = 16
5, u3 = 64
25, u4 = 256
125 D) r = 4; u1 = 4, u2 = 16
5, u3 = 64
5, u4 = 256
5
Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the
common difference; if it is geometric, find the common ratio.
7) {4n – 5}
A) Arithmetic; d = 4 B) Geometric; r =4 C) Arithmetic; d = –5 D) Neither
8) {5n2}
A) Geometric; r = 5
2B) Arithmetic; d =5 C) Geometric; r =5 D) Neither
9) 3
2
n
A) Geometric; r = 3
2B) Arithmetic; d = 3
2C) Geometric; r = 2
3D) Neither
Page 15
10) {4n2 – 2}
A) Arithmetic; d = –2 B) Geometric; r =4 C) Arithmetic; d =4 D) Neither
11) 3 – 1
5n
A) Arithmetic; d = – 1
5B) Geometric; r = – 1
5
C) Arithmetic; d =3 D) Neither
12) {4n
/
9}
A) Geometric; r = 41
/
9B) Arithmetic; d = 41
/
9
C) Geometric; r = 4 D) Neither
13) 5
,
7
,
10
,
14
,
…
A) Geometric; r = 5 B) Arithmetic; d =5 C) Geometric; r =25 D) Neither
14) 4
,
–12
,
36
,
–108
,
324
,
…
A) Geometric; r = –3 B) Geometric; r =3
C) Arithmetic; d = –16 D) Neither
15) 3, 5, 7, 11, 13, …
A) Geometric; r = 2 B) Arithmetic; d =2 C) Arithmetic; d =4 D) Neither
2 Find a Formula for a Geometric Sequence
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
1) a = 2; r = 4
A) a5 = 512; an = 2 · (4)n–1B) a5 = 512; an = 2 · (4)n
C) a5 = 2048; an = 2 · (4)nD) a5 = 2048; an = 2 · (4)n–1
2) a = 6; r = 1
4
A) a5 = 3
128 ; an = 6 · 1
4
n–1B) a5 = 3
128 ; an = 6 · 1
4
n
C) a5 = 3
512 ; an = 6 · 1
4
n–1D) a5 = 3
512 ; an = 6 · 1
4
n
3) a = 3; r = –5
A) a5 = 1875; an = 3 · (–5)n–1B) a5 = 1875; an = 3 · (–5)n
C) a5 = –375; an = 3 · (–5)n–1D) a5 = –375; an = 3 · (–5)n
4) a = –3; r = –4
A) a5 = –768; an = –3 · (–4)n–1B) a5 = –768; an = –3 · (–4)n
C) a5 = 192; an = –3 · (–4)n–1D) a5 = 192; an = –3 · (–4)n
Page 16
5) a = 4; r = 2π
A) a5 = 64π4, an = 4 · 2n–1πn–1B) a5 = 128π5, an = 4 · 2nπn
C) a5 = 64π, an = 4 · 2n–1πD) a5=4+8π
,
an= 4 + 2π(n–1)
6) a = 6
; r = 6
A) a5 = 36 6, an = 6n/2 B) a5 = 7776, an = 6n
C) a5 = 1296 6, an = 6n–1/2 D) a5 = 66, an = 6n/2–1
Find the indicated term of the geometric sequence.
7) 8th term of 1, 1
2, 1
4, …
A) a8 = 1
128 B) a8 = 1
2C) a8 = 1
64 D) a8 = 1
512
8) 7th term of 1, 2
,
4
,
…
A) a7 = 64 B) a7 =256 C) a7=128 D) a7 =4
9) 10th term of –1, 3
,
–9
,
…
A) a10 = 19,683 B) a10 = –19,683 C) a10 = –59,049 D) a10 =59,049
10) 6th term of 0.3
,
0.03
,
0.003
,
. . .
A) 0.000003 B) 0.0000003 C) 0.00003 D) 0.00000003
Find the nth term {an} of the geometric sequence. When given, r is the common ratio.
11) 2, 1, 1
2, 1
4, …
A) an = 2 1
2
n–1B) an = 2 · 2n–1C) an = 2 1
4
n–1D) an = 1
2 (2)n–1
12) 7
,
21
,
63
,
189
,
567
,
…
A) an = 7 · 3n–1B) an = 7 · 3nC) an = a1 + 3nD) an =7·3n
13) –7
,
–14
,
–28
,
–56
,
–112
,
…
A) an = –7 · 2n–1B) an = –7 · 2nC) an = a1 + 2nD) an = –7·2n
14) 4
,
–8
,
16
,
–32
,
64
,
…
A) an = 4 · (–2)n–1B) an = 4 · (–2)nC) an = a1 – 2nD) an =4·(–2)n
15) 3, 3
4, 3
16, 3
64, 3
256 , …
A) an = 3 · 1
4
n–1B) an = 3 · 1
4
nC) an = 3 · 1
4
n +1D) an = 3 · 1
16
n–1
16) 3, – 3
2, 3
4, – 3
8, 3
16, …
A) an = 3 · – 1
2
n–1B) an = 3 · – 1
2
nC) an = 3 · – 1
2
n +1D) an = 3 · – 1
4
n–1
Page 17
Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
17) a4 = –16; r = –2
A) an = 2(–2)n–1B) an = 2(–2)nC) an = 2 + (–2)n–1D) an = 128(– 1
2)n–1
18) a3 = 1
4; a6 = 1
256
A) an = 4 1
4
n–1B) an = 4(–4)n–1C) an = 1
4
1
4
n–1D) an = 1
4 + 1
4(n – 1)
Solve.
19) A new piece of equipment cost a company $70,000. Each year, for tax purposes, the company depreciates
the value by 25%.What value should the company give the equipment after 8 years?
A) $7008 B) $1 C) $9344 D) $4
20) A particular substance decays in such a way that it loses half its weight each day. How much of the
substance is left after 8 days if it starts out at 32 grams?
A) 1
8 gram B) 8 grams C) 1
4 gram D) 4 grams
21) A bicycle wheel rotates 400 times in a minute as long as the rider is pedaling. If the rider stops pedaling,
the wheel starts to slow down. Each minute it will rotate only 3/4 as many times as in the preceding
minute. How many times will the wheel rotate in the 5th minute after the rider’s feet leave the pedals?
Round your answer to the nearest unit.
A) 95 times B) 127 times C) 0 times D) 2 times
22) A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it swings
only 89% as far as on the previous swing. What is the length of the arc after 11 swings? Round your
answer to two decimal places, if necessary.
A) 15.59 inches B) 13.88 inches C) 12.35 inches D) 445 inches
23) A hockey player signs a contract with a starting salary of $890,000 per year and an annual increase of 4.5%
beginning in the second year. What will the athlete’s salary be, to the nearest dollar, in the sixth year?
A) $1,109,102 B) $1,109,927 C) $1,107,843 D) $1,111,484
24) Carla takes a job with a starting salary of $40,000 for the first year with an annual increase of 5.5%
beginning in the second year. What is Carla‘s salary, to the nearest dollar, in the sixth year?
A) $52,278 B) $53,074 C) $50,793 D) $54,612
3 Find the Sum of a Geometric Sequence
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the sum.
1) 1
7 + 3
7 + 32
7 + 33
7 + … + 3n–1
7
A) – 1
14 (1 – 3n)B)
– 1
7 (1 – 3n)C)
– 2
7 (1 – 3n)D)
1
14 (1 – 3n)
Page 18
2)
5
k=1
1
3(2)k
∑
A) 62
3B) 50
3C) 5
3D) 41
3
3)
3
k = 1
1
2
k+1
∑
A) 7
16 B) 7
8C) 11
16 D) 45
16
4)
n
k = 1
5 · 7k–1
∑
A) – 5
6 (1 – 7n)B)
– 5
6 (1 – 7n–1)C)
– 30 (1 – 7n)D)
– 5 (1 – 7n)
Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary.
5) 5 + 15 + 45 + 135 + 405 + … + 5 · 39
A) 147,620 B) 147,600 C) 147,657 D) 147,622
6) –5 – 15 – 45 – 135 – 405 – … – 5 · 310
A) –442,865 B) –442,885 C) –442,828 D) –442,863
7) 4 – 12 + 36 – 108 + 324 – … + 4 · (–3)10
A) 177,148 B) 177,146 C) 177,155 D) 177,142
8) 1
5 + 2
5 + 22
5 + 23
5 + … + 213
5
A) 3276.6 B) 3276.4 C) 6553.2 D) 6553.4
9)
5
k = 1
3(3)k
∑
A) 1089 B) 1845 C) 90 D) 117
10)
5
k = 1
3(–2)k
∑
A) –66 B) –255 C) 10 D) –18
11)
11
k = 1
1
4 · (–3)k–1
∑
A) 11,071.75 B) 11,071.25 C) 11,073.5 D) 11,070.25
Page 19
Solve.
12) A small business owner made $30,000 the first year he owned his store and made an additional 2% over
the previous year in each subsequent year. Find how much he made during his fourth year of business.
Find his total earnings during the first four years. (Round to the nearest cent, if necessary.)
A) $31,836.24; $123,648.24 B) $0.24; $30,612.24
C) $51,840.00; $161,040.00 D) $62,208.00; $223,248.00
13) As Sunee improves her algebra skills, she takes 0.8 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.)
A) 20 min; 168 min B) 16 min; 168 min C) 20 min; 148 min D) 16 min; 148 min
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) Initially, a pendulum swings through an arc of 3 feet. On each successive swing, the length of the arc is 0.8
of the previous length. After 10 swings, what total length will the pendulum have swung (to the nearest
tenth of a foot)?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
15)
J
oytown has a present population of 40,000 and the population is increasing by 2.5% each year. How long
will it take for the population to double? Round your answer to the nearest year.
A) 28 years B) 40 years C) 29 years D) 41 years
4 Determine Whether a Geometric Series Converges or Diverges
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
1) 5 + 5
3 + 5
9 + 5
27 + . . .
A) Converges; 15
2B) Converges; 20
3C) Converges; 5
3D) Diverges
2) 1 – 1
4 + 1
16 – 1
64 + . . .
A) Converges; 4
5B) Converges; 3
4C) Converges; – 1
4D) Diverges
3) 48 + 12 + 3 + 3
4 + . . .
A) Converges; 64 B) Converges; –16 C) Converges; 63 D) Diverges
4) 3 + 1 + 1
3 + …
A) Converges; 9
2B) Converges; 3 C) Converges; 1 D) Diverges
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5) 2 – 1
2 + 1
8 – …
A) Converges; 8
5B) Converges; 2 C) Converges; – 1
2D) Diverges
6) 24 + 12 + 6 + …
A) Converges; 48 B) Converges; –24 C) Converges; 42 D) Diverges
7) –20 – 5 – 5
4 – …
A) Converges; – 80
3B) Converges; 20
3C) Converges; – 105
4D) Diverges
8) 10 + 14 + 16 + 18 + …
A) Converges; 29 B) Converges; 40 C) Converges; ∞D) Diverges
9)
∞
k = 1
1
8 · 5k – 1
∑
A) Converges; 5
8B) Converges; 25
4C) Converges; ∞D) Diverges
10)
∞
k=1
4 2
3
k–1
∑
A) Converges; 12 B) Converges; 16 C) Converges; 4 D) Diverges
Solve.
11) A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it swings
only 60% as far as on the previous swing. How far will it swing altogether before coming to a complete
stop?
A) 125 inches B) 83 inches C) 63 inches D) 167 inches
12) A ball is dropped from a height of 25 feet. Each time it strikes the ground, it bounces up to 0.7 of the
previous height. The total distance the ball has traveled before the second bounce is 25 + 2(25 · 0.7) feet,
and the total distance the ball has traveled before bounce n + 1 is
25 +
n
k=1
50 0.7k
∑feet.
Use facts about infinite geometric series to calculate the total distance the ball has traveled by the time it
has stopped bouncing.
A) 141 2
3 feet B) 141 2
5 feet C) 140 1
3 feet D) 144 3
5 feet
13) A ping–pong ball is dropped from a height of 9 ft and always rebounds 1
3 of the distance fallen. Find the
total sum of the rebound heights of the ball.
A) 4.5 ft B) 13.5 ft C) 6 ft D) 3 ft
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14) After being struck with a hammer, a gong vibrates 20 vibrations in the first second and in each second
thereafter makes 2
3 as many vibrations as in the previous second. Find how many vibrations the gong
makes before it stops vibrating.
A) 60 vibrations B) 30 vibrations C) 70 vibrations D) 25 vibrations
13.4 Mathematical Induction
1 Prove Statements Using Mathematical Induction
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
1) 2 + 7 + 12 + … + (5n – 3) = n
2(5n – 1)
2) 7 + 14 + 21 + … + 7n = 7n(n + 1)
2
3) 1 + 5 + 52 + … + 5n – 1 = 5n – 1
4
4) n2 – n + 2 is divisible by 2
5) 1
2 + 1
4 + 1
8 + 1
16 + … + 1
2n = 1 – 1
2n
6) Use the Principle of Mathematical Induction to show that the statement “5 is a factor of 7n – 2n” is true for
all natural numbers. (Hint: 7k+1 – 2k+1 = 7(7k – 2k) + 5 · 2k)
7) Show that the formula
2 + 4 + 6 + 8 + … + 2n = n2 + n + 3
obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for
some natural number k, it is also true for the next natural number k + 1. Then show that the formula is
false for n = 1.
8) Show that the statement “n2 – n + 3 is a prime number” is true for n = 1, but is not true for n = 3.
9) 6 + 12 + 18 + … + 6n = 3n(n + 1)
10) 5 + 5 · 1
6 + 5 · 1
6
2 + … + 5 · 1
6
n – 1 =
51
– 1
6
n
1 – 1
6
11) 12 + 42 + 72 + . . . + (3n – 2)2 = n(6n2 – 3n – 1)
2
12) 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) = n(n +1)(n +2)
3
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13) 1 – 1
2 1
– 1
3 . . . 1 – 1
n + 1 = 1
n + 1
14) 1 · 4 + 2 · 4 + 3 · 4 + . . . + 4n = 4n(n + 1)
2
15) 46n = 46n
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve.
16) Let P(n) represent the statement:
–3 + 3 + 9 + … + (6n – 9) = 3n2 – 6n
In the proof that P(n) is true for all integers n, n ≥ 1, what term must be added to both sides of P(k) to
show P(k +1) follows from P(k)?
A) 6k – 3 B) P(k +1) C) 6k –9 D) 6k +3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
17) Let P(n) represent the statement:
–2 + 4 + 10 + … + (6n – 8) = 3n2 – 5n
In the proof that P(n) is true for all integers n, n ≥ 1, what term must be added to both sides of P(k) to
show P(k +1) follows from P(k)?
13.5 The Binomial Theorem
1 Evaluate a Binomial Coefficient
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the expression.
1)
9
3
A) 84 B) 60,480 C) 42 D) 3
2) 8
2
A) 28 B) 8 C) 0 D) 1
3) 10
5
A) 252 B) 504 C) 30,240 D) 126
4)
188
2
A) 17,578 B) 35,156 C) 188!
186! D) 186
5)
10
0
A) 1 B) 3,628,800 C) 2 D) 0
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6) 10
1
A) 10 B) 1 C) 10
9D) 10!
7)
4
4
A) 1 B) 24 C) 2 D) 0
2 Use the Binomial Theorem
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Expand the expression using the Binomial Theorem.
1) (x – 1)6
A) x6 – 6x5 +15x4 – 20x3 + 15x2 – 6x + 1B)x
6 – 6x5 – 15x4 – 20x3 – 15x2 – 6x – 6
C) x6 – 6x5 +30x4 – 120x3 + 30x2 – 6x + 1D)x
6 – 6x5 – 30x4 – 120x3 – 360x2 – 720x + 720
2) (x – 7)5
A) x5 – 35x4 + 490x3 – 3430x2 + 12,005x – 16,807 B) x5 – 35x4 + 490x3 – 3430x2 + 12,005x – 7
C) x5 – 35x4 + 980x3 – 6860x2 + 12,005x – 16,807 D) x5 – 35x4 + 980x3 – 6860x2 + 12,005x – 7
3) (4x + 2)3
A) 64x3 + 96x2 + 48x + 8 B) 64x3 + 96x2 + 96x + 8
C) 16x6 + 8x3 + 64 D) 16x2 + 16x + 4
4) (3x – 1)4
A) 81x4 – 108x3 + 54x2 – 12x + 1B)
–81x4 + 108 x3 + 54x2 + 12x + 1
C) (9x2 – 3x + 1)4D) 81x3 – 108x2 + 54x – 12
5) (4x – 2)5
A) 1024x5 – 2560x4 + 2560x3 – 1280x2 + 320x – 32 B) 1024x5 – 512x4 + 256x3 – 128x2 + 64x – 32
C) (16x2 – 16x + 4)5D) 1024x5 + 320x4 – 1280x3 – 1280x2 + 320x – 32
6) (3x2 + 2)3
A) 27x6 + 54x4 + 36x2 + 8B)(9x
4 + 12x2 + 4)3
C) 27x3 + 54x2 + 36x + 8 D) 81x8 + 27x6 + 54x4 + 36x2 + 8
7) (2x – 5y)3
A) 8x3 – 60x2y + 150xy2 – 125y3B) 8x3 – 20x2y + 50xy2 – 125y3
C) 4x3y – 10x2y2 + 25xy3D) 4x3y – 20x2y2 + 25xy3
8) (x2 – 2y)4
A) x8 – 8x6y + 24x4y2 – 32x2y3 + 16y4B) x8 – 2x6y + 24x4y2 – 16x2y3 + 16y4
C) x4 – 8x3y + 24x2y2 – 32xy3 + 16y4D) x8 – 8x6y + 24x4y2 + 8x2y3 + 16y4
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9) (x2 + 4y)4
A) x8 + 16x6y + 96x4y2 + 256x2y3 + 256y4B) x6 + 12x5y + 96x4y2 + 192x2y3 + 256y4
C) x6 + 16x5y +24x4y2 + 16x2y3 + 4y4D) x8 + 12x6y + 96x4y2 + 192x2y3 + 256y4
10) (2x + y)6
A) 64x6 + 192x5y + 240x4 y2 + 160x3 y3 + 60x2y4 + 12xy5 + y6
B) 2x6 + 12x5y + 30x4 y2 + 40x3y3 + 12xy5 + y6
C) 64x6 + 192x5y + 480x4y2 + 960x3y3 + 1440x2y4 + 12xy5 + y6
D) 64x6 + 192x5y + 240x4y2 + 160x3y3 + 240x2y4 + 192xy5 + 64y6
11) (w – s)6
A) w6 – 6w5s + 15w4s2 – 20w3s3 + 15w2s4 – 6ws5 + s6
B) w6 – s6
C) w6 – 8w5s + 17w4s2 – 22w3s3 + 17w2s4 – 8ws5 + s6
D) w6 – 6w5s – 30w4s2 + 120w3s3 + 360w2s4 – 720ws5 – 720s6
12) (g – 2h)3
A) g3 – 6g2h + 12gh2 – 8h3B) g3 – 3g2h + 6gh2 – 2h3
C) g3 – 6h2g + 12hg2 – 8h3D) g3 – 8h3
13) x – 3
x
4
A) x4 – 12x5/2 + 54x – 108
x + 81
x2B) x4 – 12x3/2 + 54x – 108
x + 81
x2
C) x4 – 81
x2D) x4 + 81
x2
14) ( x + 3)4
A) x2 + 43
x3/2 + 18x + 12 3x1/2 + 9B)x
2 + 23x3/2 + 9x + 63x1/2 + 9
C) x2 + 43
x1/2 + 18x + 12 3x1/2 + 81 D) x2 + 9
15) (ax + by)5
A) a5x5 + 5a4bx4y + 10a3b2x3y2 + 10a2b3x2y3 + 5ab4xy4 + b5y5
B) a5x5 + 5a3b2x3y2 + 5ab4xy4 + b5y5
C) a5x5 + 5abx4y + 10a3b2x3y2 + 10a2b3x2y3 + 5abxy4 + b5y5
D) a5x5 + b5y5
Use the Binomial Theorem to find the indicated coefficient or term.
16) The coefficient of x in the expansion of (9x + 4)3
A) 432 B) 972 C) 16 D) 864
17) The coefficient of x in the expansion of (2x + 4)5
A) 2560 B) 1280 C) 5120 D) 640
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18) The coefficient of 1
x in the expansion of 2x + 1
x
3
A) 6 B) 2 C) 9 D) 3
19) The coefficient of x4 in the expansion of (3x + 4)6
A) 19,440 B) 34,560 C) 11,664 D) 23,328
20) The coefficient of x8 in the expansion of (x2 – 3)7
A) –945 B) 945 C) 2835 D) –2835
21) The 3rd term in the expansion of (7x + 6)3
A) 756x B) 882x2C) 36 D) 1512x
22) The 5th term in the expansion of (4x + 3)5
A) 1620x B) 2160x2C) 1215 D) 540x
23) The 10th term in the expansion of (2x – 3y)12
A) –34,642,080x3y9B) 11,547,360x3y10 C) 11,547,360x9y3D) –17,321,040x9y3
24) The 11th term in the expansion of (x – 3y)13
A) 16,888,014x3y10 B) –5,629,338x3y11 C) –5,629,338x10y3D) 16,888,014x10y3
25) The 5th term in the expansion of (2x – 2y)12
A) 2,027,520x8y4B) –1,013,760x8y5C) –1,013,760x4y8D) 1,013,760x4y8
26) The 5th term in the expansion of (x – 3y)9
A) 10,206x5y4B) –3402x5y5C) –3402x4y5D) 10,206x4y5
Solve.
27) Use the Binomial Theorem to approximate (1.01)5 = (1 + 10–2)5 to 7 decimal places.
A) 1.0510101 B) 1.050101 C) 1.0550101 D) 1.0550505
28) Use the Binomial Theorem to approximate (0.98)6 = (1 – 2(10–2))6to 5 decimal places.
A) 0.88584 B) 0.85841 C) 0.88854 D) 0.88548
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Ch. 13 Sequences; Induction; the Binomial Theorem
Answer Key
13.1 Sequences
1 Write the First Several Terms of a Sequence
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13.2 Arithmetic Sequences
1 Determine if a Sequence is Arithmetic
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13.3 Geometric Sequences; Geometric Series
1 Determine if a Sequence is Geometric
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13.4 Mathematical Induction
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13.5 The Binomial Theorem
1 Evaluate a Binomial Coefficient
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