Deck 10: Additional Topics in Algebra

Question 1

Write out the first four terms in the sequence.

A)
B)
C)
D)

Answer 1

Question 2

Expand and evaluate the series.

Answer 2

140

Question 3

Use the following to answer questions :
14, 20, 26, 32, 38, . . .
Predict the next term for the sequence.

A) 55
B) 56
C) 57
D) 58

Answer 3

57

Question 4

Find the indicated term for the sequence. a_n = ; a₁₉.

A)
B)
C)
D)

Question 5

Write out the first four terms in the sequence.

Question 6

Simplify the factorial expression.

A)
B) 5
C) 720
D) 5760

Question 7

Use the following to answer questions :
10, 100, 1000, 10,000, 100,000, . . .
Find the general term for the sequence.

A) a_n = 10n
B) a_n = 10ⁿ
C) a_n = n¹⁰
D) a_n = 10^{n - 1}

Question 8

Use the following to answer questions :
a_n = (-1)ⁿ (n + 1)
Find the 20th term.

A) 21
B) -21
C) 20
D) -20

Question 9

Find the first five terms of the recursive sequence.

Question 10

Simplify the factorial expression.

A) 495
B) 55
C)
D)

Question 11

Find the indicated term for the sequence.
a_n = ; a₁₀.

Question 12

Expand and evaluate the series.

A) -3
B) -4
C) -5
D) -6

Question 13

Use the following to answer questions :
14, 20, 26, 32, 38, . . .
Find the general term for the sequence.

A) a_n = n + 11
B) a_n = 5n + 7
C) a_n = -4n + 16
D) a_n = 4n + 8

Question 14

Simplify the factorial expression.

Question 15

Find the partial sum S₄ for the sequence. a_n = -4n + 5

A) -21
B) -20
C) -19
D) -18

Question 16

(a) Find the first four terms for the nth term given.
(b) Find the 9th term.
a_n = 9n + 4

Question 17

Write the sum using sigma notation. Answers are not necessarily unique.
10 + 20 + 30 + 40 + 50 + 60

Question 18

Simplify the factorial expression.

Question 19

Use the following to answer questions :
a_n = (-1)ⁿ (n + 1)
Find the first four terms and write them as a list.

A) 1, -2, 3, -4
B) -1, 2, -3, 4
C) 2, -3, 4, -5
D) -2, 3, -4, 5

Question 20

Use the following to answer questions :
10, 100, 1000, 10,000, 100,000, . . .
Predict the next term for the sequence.

Question 21

Find the number of terms in the series and then find the sum using the alternate formula. 11 + 19 + 27 + 35 + 43 + . . . + 163

A) n = 19; S₂₀ = 1583
B) n = 19; S₂₀ = 1591
C) n = 20; S₂₀ = 1740
D) n = 21; S₂₀ = 1911

Question 22

Use the following to answer questions :
Kelly earns $8.80 per hour. Each year she receives a $0.80 hourly raise.
How much will Angie make the fifth year if she works 8 hr per day for 250 working days?

A) $19,400.00
B) $20,800.00
C) $22,200.00
D) $23,600.00

Question 23

(a) Identify the first term a₁ of the arithmetic sequence.
(b) Identify the common difference d.
(c) Write an expression for the general term a_n.
(d) Use the general term to find the 12th term of the sequence.
11, 14, 17, 20, 23, . . .

Question 24

Evaluate the sum. Use the summation properties from Section 8.1.

A) 112
B) 110
C) 108
D) 106

Question 25

Find the number of terms n in an arithmetic sequence if a₁ = 13, a_n = -95, and d = -6

A) 17
B) 18
C) 19
D) 20

Question 26

Determine if the sequence given is arithmetic. If yes, name the common difference. -3, -6, -12, -18, -24, . . .

A) arithmetic; d = 2
B) arithmetic; d = 3
C) arithmetic; d = -3
D) not arithmetic

Question 27

Use the following to answer questions :
Kelly earns $8.80 per hour. Each year she receives a $0.80 hourly raise.
List Felicia's hourly wage for the first 5 years.

Question 28

Write the sum using sigma notation. Answers are not necessarily unique. 1 - 4 + 9 - 16 + 25

A)
B)
C)
D)

Question 29

Write the first four terms of the arithmetic sequence with the given first term and common difference. a₁ = 10, d = -2

A) -2, -12, -22, -32
B) -2, 8, 18, 28
C) 10, 12, 14, 16
D) 10, 8, 6, 4

Question 30

Write the series in sigma notation. Simplify the series (if possible) before you begin.

A)
B)
C)
D)

Question 31

Write the series in sigma notation. Simplify the series (if possible) before you begin.

Question 32

Find the common difference d and the value of a₁ given a₅ = -24, and a₁₆ = -123.

Question 33

Use the alternative formula for the nth partial sum to compute S₁₅ for the sequence.
-18 + (-15.5) + (-13) + (-10.5) + (-8) + . . .

Question 34

Find the sum of the first 20 multiples of 6.

Question 35

Find a₁₂ for an arithmetic sequence with a₁ = 14 and d = -2.

A) -6
B) -8
C) -10
D) -12

Question 36

Determine if the sequence given is arithmetic. If yes, name the common difference. 4, 6, 8, 10, 12, . . .

A) arithmetic; d = 2
B) arithmetic; d = -2
C) arithmetic; d = 4
D) not arithmetic

Question 37

Write the first four terms of the arithmetic sequence with the given first term and common difference.
a₁ = 3, d = 3

Question 38

For the given general term a_n, write the sum S₇ using sigma notation.
a_n = n + 9

Question 39

Jim deposits $15,000 in an account on January 1. Starting on January 2, he withdraws $100 each morning. Find the amount in his account on the evening of January 27. (Hint: For a₁ = 15,000, the amount remaining 26 days later will be what term of the sequence?)

A) $12,700
B) $12,600
C) $12500
D) $12,400

Question 40

Evaluate the sum.

Question 41

Use the following to answer questions :
a_n = 7n + 5
Find a₄.

A) 22
B) 29
C) 36
D) 43

Question 42

Find the indicated sum. Use the summation properties from Section 8.1. Round your answer to 2 decimal places.

A) 3.96
B) 3.98
C) 4.00
D) 4.02

Question 43

Determine if the sequence given is geometric. If yes, name the common ratio. 9, 4, -1, -6, -11, . . .

A) geometric; r = -5
B) geometric; r = 5
C) geometric; r = 9
D) not geometric

Question 44

Use the following to answer questions :
A biologist finds that the population of a certain type of bacteria doubles each half-hour. An initial culture has 90 bacteria.
How long will it take for the number of bacteria to reach 245,760?

A) 5.5 hours
B) 6 hours
C) 6.5 hours
D) 7 hours

Question 45

Determine if the sequence given is geometric. If yes, name the common ratio. 4, -12, 36, -108, 324, . . .

A) geometric; r = 3
B) geometric; r = 4
C) geometric; r = -3
D) not geometric

Question 46

Use the following to answer questions :
A biologist finds that the population of a certain type of bacteria doubles each half-hour. An initial culture has 90 bacteria.
What is the population after 5 hours?

A) 100
B) 200
C) 640
D) 20,480

Question 47

Find the number of terms n in the sequence.

A) 9
B) 10
C) 11
D) 12

Question 48

Find a₆ for a geometric sequence with a₁ = 3 and r = .

A)
B) 27
C)
D) 81

Question 49

Find the number of terms n in the sequence.
a₁ = 3, a_n = 192, and r = 2.

Question 50

Use the following to answer questions :
a_n = 7n + 5
Find a₅.

A) 7
B) 11
C) 15
D) 19

Question 51

Determine whether the infinite geometric series has a finite sum. If so, find the limiting value. 112 + 56 + 28 + . . .

A) S_∞ = 224
B) S_∞ = -224
C) S_∞ = 56
D) no finite sum

Question 52

Use the following to answer questions :
A geometric sequence has terms a₃ = 576 and a₆ = 243.
Find the common ratio r.

A)
B)
C)
D)

Question 53

Write the first four terms of the geometric sequence given a₁ = and r = .

A)
B)
C)
D)

Question 54

Use the following to answer questions :
A geometric sequence has terms a₃ = 576 and a₆ = 243.
Find the value of a₁.

A) 256
B) 1024
C) 3072
D) 4096

Question 55

Write the first four terms of the geometric sequence given a₁ = 2 and r = -4.

Question 56

Determine whether the infinite geometric series has a finite sum. If so, find the limiting value. 9 + 27 + 81 + 243 + 729 + . . .

A) S_∞ =
B) S_∞ =
C) S_∞ =
D) no finite sum

Question 57

(a) Identify the first term a₁ of the geometric sequence.
(b) Identify the common ratio r.
(c) Write an expression for the nth term a_n = a₁r^n-1.
(d) Use the expression for the nth term to find the 10th term.

Question 58

Find the sum S₁₁ for a geometric sequence with a₁ = 9 and r = -2.

Question 59

Use the following to answer questions :
a_n = 7n + 5
Find a_k.

A) a_k = 7k + 3
B) a_k = 7n + 3
C) a_k = (7n + 3)k
D) a_k = (7n + 3) + k

Question 60

Determine whether the infinite geometric series has a finite sum. If so, find the limiting value.

Question 61

Use the following to answer questions :
a_n = 7n + 5
Find a_{k + 1}.

A) a_{k + 1} = 7k - 1
B) a_{k + 1} = 7k + 3
C) a_{k + 1} = 7k + 4
D) a_{k + 1} = 7k + 5

Question 62

Use mathematical induction to prove the indicated sum formula is true for all natural numbers n.
7 + 14 + 21 + 28 + . . . + 7n = ;
a_n = 7n, S_n =

Question 63

Find the value of ₇P₁ either by computing r factors of n! or by using the formula .

A) 0
B) 1
C) 7
D) 5040

Question 64

Use the following to answer questions :
A teacher is going to assign each student a 3-digit code using the digits 0 through 3.
How many codes are possible if repetitions are allowed?

A) 18
B) 120
C) 216
D) 125

Question 65

For S_n = n(5n - 3), find S₄, S₅, S_k, and S_{k + 1}.

Question 66

Use mathematical induction to prove that n² - 13n is divisible by 2.

Question 67

A Scrabble player has the six letters (A, A, E, E, N, N) in her rack. How many distinguishable, six-letter permutations can be formed?

A) 90
B) 240
C) 360
D) 720

Question 68

Alex has 6 shirts, 6 pairs of pants, and 5 pairs of shoes. How many outfits can he wear?

Question 69

How many different ways can 6 people stand in line at a cash register?

Question 70

How many distinguishable permutations can be formed from the letters of the word assassin?

Question 71

A coin is flipped three times. Draw a tree diagram illustrating all possible outcomes. How many elements are in the sample space?

Question 72

Assuming the triplets are identical and cannot be told apart, how many distinguishable photographs can be taken of a family of 5, if they stand in a single row and there is one set of triplets?

A) 60
B) 40
C) 20
D) 120

Question 73

Use mathematical induction to prove the indicated sum formula is true for all natural numbers n.
(-3) + (-1) + 1 + 3 + . . . + (2n - 5);
a_n = 2n - 5, S_n = n(n - 4)

Question 74

Find the value of ₅P₃ either by computing r factors of n! or by using the formula .

Question 75

Use the following to answer questions :
A teacher is going to assign each student a 3-digit code using the digits 0 through 3.
How many codes are possible if no repetitions are allowed?

A) 24
B) 125
C) 60
D) 64

Question 76

Use the following to answer questions :
A teacher is going to assign each student a 3-digit code using the digits 0 through 3.
How many codes are possible if repetitions are not allowed and the code must be less than 200 and divisible by 10?

A) 2
B) 24
C) 12
D) 3

Question 77

A lock uses the numbers 0 through 12 for a four-number combination. How many different combinations are possible?

A) 48
B) 20,736
C) 28,561
D) 248,832

Question 78

A club must elect a president, vice-president, and treasurer. In how many ways can the positions be filled if the club has 12 members?

A) 36
B) 220
C) 1320
D) 1728

Question 79

Assuming the twins are identical and cannot be told apart, how many distinguishable photographs can be taken of a family of 4, if they stand in a single row and there is one set of twins?

Question 80

Use mathematical induction to prove the statement is true for all natural numbers n.
3ⁿ ≥ n + 2.