Free Printable Discrete Math: Review of Units 1-3

1.

Which sequence(s) represent an arithmetic sequence?

a)

a and e

b)

a, b, and e

c)

b, c, and d

d)

b

2.

Is this an arithmetic sequence?

a)

yes

b)

maybe

c)

sometimes

d)

no

3.

Write the explicit rule for this situation.

a)

f(n) = 35 + 3(n − 1)

b)

f(n) = 35 −3(n − 1)

c)

f(n) = 3 + 35(n − 1)

d)

f(n) = −3 + 35(n − 1)

4.

What is the common difference?

a)

−16

b)

84

c)

16

d)

0

5.

Write the recursive formula for 4, 8, 12...

a)

a_n=a_n-1-7

b)

a_n=a_n-1-4

c)

a_n=a_n-1+3

d)

a_n=a_n-1+4

6.

Given the sequence:
25, 21, 17, 13,...
Write the explicit equation that models the sequence. (Hint: Simplify your equation.)

a)

a_n = 4n + 29

b)

a_n = -4n + 25

c)

a_n= -4n + 29

d)

a_n = 4n + 25

7.

Find the 32nd term of this sequence.
9, 4, -1, -6, -11, ...

a)

81

b)

-34

c)

-146

d)

-225

8.

Given an arithmetic sequence where a₃₂ = 109 and a₄₆ = 193, find a₃₁₁.

a)

295

b)

-77

c)

1860

d)

1783

9.

What does r represent?

a)

Radical

b)

Common difference

c)

Common ratio

10.

Is the sequence -432, 72, -12, 2,...geometric?

a)

yes

b)

no

11.

Write the explicit formula for the geometric sequence.

a)

a_n = 4(3)^n-1

b)

a_n = 3(4)^n-1

c)

a_n = -4(3)^n-1

d)

a_n = 3(-4)^n-1

12.

What is the 14th term of the geometric sequence?

a)

14,348,907

b)

4,782,969

c)

1,594,323

d)

45

13.

The rule for finding a_n of a geometric sequence is

a_n = 10(2)^n-1_{What is the common ratio(r) of the sequence?}

a)

²

b)

¹⁰

c)

²⁰

d)

2ⁿ

14.

A geometric sequence has a common ration(r) of 3 and the 8th term (a₈) is 13,122. What is the first term of the sequence (a₁)?

a)

3

b)

6

c)

18

d)

2

15.

Daisy received $2 on her first birthday. Each year, the amount of money she received tripled. How much money will she receive on her 5^th birthday?

a)

$10

b)

$300

c)

$162

d)

$416

16.

Write the explicit formula for the geometric sequence.

a)

a_n = 162(2/3)^n-1

b)

a_n = (2/3)(162)^n-1

c)

a_n = 162(3/2)^n-1

d)

a_n = 162 + (n-1)(-54)

17.

Which formula is used to find the recursive equation for geometric sequences?

a)

a_n = a_n-1 + d

b)

a_n = a_n-1 * r

c)

a_n = a₁ + d(n-1)

d)

a_n = a₁ * r^n-1

18.

Write the recursive rule for the geometric sequence:
64, 16, 4,...

a)

a₁ = 64; a_n = a_n-1 - ¼

b)

a₁ = 4; a_n = 4(a_n-1)

c)

a₁ = 64; a_n = 4(a_n-1)

d)

a₁ = 64; a_n = 1/4(a_n-1)

19.

Find the eighth term of a geometric sequence for which a₃ = 224 and r = -½.

a)

-28

b)

-112

c)

56

d)

-7

20.

Given a geometric sequence where a₃ = 320 and a₆ = 20480, find a₈

a)

4

b)

20

c)

16,384

d)

327,680

21.

Evaluate the arithmetic series described:
2 + (-2) + (-6) + (-10)..., S₁₀

a)

-160

b)

-328

c)

-320

d)

-336

22.

Evaluate the arithmetic series described:
a₁ = 12, a_n = 64, Find S₁₄

a)

1082

b)

532

c)

541

d)

1064

23.

Find the sum of the 1st 20 terms: 2+5+8+......

a)

1220

b)

610

c)

600

d)

612

24.

Find the sum: 11 + 15 + 19 + .......+ 83

a)

19

b)

83

c)

893

d)

624

25.

Find S_n for the given series:
a₁=22
d=-8
n=21

a)

1911

b)

-1449

c)

-2436

d)

-1218

26.

Find how many terms are in the series below:
7+13+19+25...=1044

a)

16

b)

17

c)

18

d)

19

27.

Find the missing terms of the arithmetic sequence:
145, ___, ___, ___, 205

a)

160, 175, 190

b)

165, 185, 195

c)

155, 165, 175

d)

190, 175, 160

28.

Find the sum of the first 25 terms of the series: 6+14+22+30+...

a)

450

b)

2550

c)

2650

d)

550

29.

The local power company plans to raise rates to cover the increased cost of producing power. They determine the costs will increase by 4.2% per year for the next 10 years. If you paid a total of $1031.60 for power last year, what would you expect to pay for power over the next 10 years?
(Hint: Find a₁, a₁₀, and r)

a)

$13,020.97

b)

$10,316.00

c)

$2,704.97

d)

$12,220.97

30.

Find the sum of the finite geometric series:

a)

88419

b)

78124

c)

91846

d)

92234

31.

Find the sum of the infinite geometric series, if it exists. $\sum_{n=1}^{\infty}8\left(\frac{1}{5}\right)^{n-1}$

a)

8

b)

8/5

c)

10

d)

Does not exist

32.

Evaluate.

a)

290

b)

278

c)

250

d)

295

33.

Is this sum divergent or convergent?

a)

Divergent, because the sum adds up to infinity.

b)

Convergent, because the sum adds up to about 1/4

34.

Is this sum divergent or convergent?

a)

Convergent, because the sum adds up to a single value (0.5)

b)

Divergent, because the sum adds up to infinity.

35.

What is the summation notation for the series?

7 + 11 + 15 + ... + 203 + 207

a)

b)

c)

d)

36.

Is the sequence convergent or divergent?

a)

Convergent

b)

37.

Evaluate the infinite geometric series:

1 + 1/5 + 1/25 + ...

a)

5/4

b)

9/5

c)

5/6

d)

65/27

38.

Find the sum of the infinite geometric series, if it exists. $\frac{1}{2}-\frac{5}{3}+\frac{50}{9}-\frac{500}{27}+...$

a)

205

b)

19.5

c)

108.75

d)

Does Not Exist

39.

A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern.

If the theater has 20 rows of seats, how many seats are in the 20th row of the theater?

a)

172 seats

b)

180 seats

c)

188 seats

d)

212 seats

40.

For the month of April, your parents offer to pay you $0.10 for the first day, $0.20 for the second day, $0.40 for the third day, $0.80 for the fourth day, and so on. How much money would they give you on the last (30th) day of April?

(a)

41.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²
On the basis of this assumption,
[The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²]
What must we show?

a)

The statement is true for n = 1:
2x1 − 1 = 1²

b)

The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

c)

The statement is true for n = k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

42.

$\sum_{k=1}^{\infty}\left(\frac{2}{5}\right)^k$ Find the sum

a)

$\frac{2}{5}$

b)

$\frac{2}{3}$

c)

0

d)

$\infty$

43.

What is row 5 of Pascal's Triangle?

a)

1, 2, 1

b)

1, 3, 3, 1

c)

1, 4, 9, 4, 1

d)

1, 5, 10, 10, 5, 1

44.

Which of these is the first step in mathematical induction?

a)

Prove the statement is true for the first element in the set.

b)

Show that if the statement is true for the first k elements, then it is true for the (k+1)st case.

c)

Prove that the problem you are working on is the base to all proofs.

d)

None of these are correct.

45.

Expand $\left(2x+4y\right)^3$

a)

$8x^3+48x^2y+96xy^2+64y^3$

b)

$40x^3+160x^2y+320xy^2+320y^3$

c)

$32x^3+48x^2y+96xy^2+64y^3$

d)

$8x^3+16x^2y+32xy^2+64y^3$

46.

Select the correct expansion of

(1 + 4x)³

a)

1 + 12x + 12x² + 4x³

b)

1 + 12x + 48x² + 64x³

c)

1 + 3x + 3x² + x³

d)

1 + 4x + 16x² + 64x³

47.

Find B-A

a)

-3 1
7 -1

b)

1 0
1 -4

c)

4 0
-6 0

d)

not possible because rows do not match columns

48.

What are the dimensions of this matrix?

a)

2 x 3

b)

3 x 2

c)

6 x 1

d)

1 x 6

49.

Multiply

a)

20 15 -10
30 -5 0

b)

-20 15 -10
30 -5 0

c)

1 8 3
11 4 5

d)

20 -15 10
-30 5 0

50.

Find A+B

a)

8 10
8 1

b)

12 25
7 1

c)

10 30
0 1

d)

can't add two together because the rows dont match columns

51.

Find the values of x, y and z.

a)

w = 13, x = 4, y = 7, z = 23

b)

w = 13, x = -4, y = 11, z = 23

c)

w = 13, x = 4, y = 11, z = 23

d)

w = 13, x = 4, y = 11, z = 17

52.

How many rows are in a 7 x 3 matrix?

a)

7

b)

3

c)

21

d)

10

53.

You can multiply a 2X3 matrix by which matrix?

a)

2X2

b)

2X12

c)

3X12

d)

2X3

54.

Find the product of the two matrices.

a)

undefined

b)

-3 16
17 12
8 -10

c)

3 12
-15 -8
-1 -3

55.

In Matrix R (pictured) what element is a₂₃?

a)

8

b)

9.01

c)

6

d)

1

56.

find: B*A

a)

47 30
9 5

b)

17 15
42 35

c)

10 25
7 1

d)

not possible because rows do not match columns

57.

These matrices are being multiplied. Determine the dimension/size of the new matrix.

a)

Can't multiply them

b)

4 x 2

c)

3 x 3

d)

4 x 3

58.

Evaluate the det (B).

a)

7

b)

-7

c)

23

d)

-23

59.

Determine if the matrix has an inverse, if so find it.

a)

No

b)

0 1
-1 2

c)

-1 3
1 -2

d)

2 -1
-1 1

60.

Determine if the matrix has an inverse, if so find it.

a)

No

b)

0 1
-1 2

c)

-1 3
1 -2

d)

2 -1
-1 1

61.

Solve the system of equations using matrices:

a)

( -1, 6, 1)

b)

( 1, 6, -1)

c)

None of these

d)

( -6, -1, 1)

62.

Solve using matrices:

y = 7x + 9

2y + 2x = -18

a)

(5, 0)

b)

(9, 18)

c)

(-2, -5)

d)

(1, 16)

63.

Amy sold 10 packages of sugar cookies and 2 packages of oatmeal cookies for $56. Adam sold 9 packages of sugar cookies and 3 packages of oatmeal cookies for $60. What is the price of 1 pack of sugar cookies and 1 pack of oatmeal cookies?

Set up your equations, then solve using matrices.

a)

Sugar Cookies $5

Oatmeal Cookies $3

b)

Sugar Cookies $4

Oatmeal Cookies $8

c)

Sugar Cookies $2

Oatmeal Cookies $6

d)

Sugar Cookies $3

Oatmeal Cookies $6

64.

The Super X sells three kinds of sandwiches – Super X Original, Italian Special, and Barbecue Beef. The Super X clerk observed what people purchased depended on what they had ordered on the previous visit. He conducted a survey and found that of the students who ordered the Original on their last visit, 20% ordered it again the next time, whereas 25% switched to Italian and 55% switched to Barbecue Beef. Of the students who ordered the Italian sandwich the last time, 35% did so again the next time, but 45% switched to the Original and 20% switched to the Barbecue Beef. Of the students who got the Barbecue Beef the last time, 55% ordered it the next time, 20% switched to the Original, and 25% switched to Italian. Write a transition matrix

a)

b)

c)

65.

The Super X sells three kinds of sandwiches – Super X Original, Italian Special, and Barbecue Beef. The Super X clerk observed what people purchased depended on what they had ordered on the previous visit. He conducted a survey and found that of the students who ordered the Original on their last visit, 20% ordered it again the next time, whereas 25% switched to Italian and 55% switched to Barbecue Beef. Of the students who ordered the Italian sandwich the last time, 35% did so again the next time, but 45% switched to the Original and 20% switched to the Barbecue Beef. Of the students who got the Barbecue Beef the last time, 55% ordered it the next time, 20% switched to the Original, and 25% switched to Italian. Write an initial matrix for a person who buys an Italian Sandwich on Monday.

a)

[.45 .35 .2]

b)

[0 1 0]

c)

[.35 .45 .2]

d)

[1 0 0 ]

66.

The Super X sells three kinds of sandwiches – Super X Original, Italian Special, and Barbecue Beef. The Super X clerk observed what people purchased depended on what they had ordered on the previous visit. He conducted a survey and found that of the students who ordered the Original on their last visit, 20% ordered it again the next time, whereas 25% switched to Italian and 55% switched to Barbecue Beef. Of the students who ordered the Italian sandwich the last time, 35% did so again the next time, but 45% switched to the Original and 20% switched to the Barbecue Beef. Of the students who got the Barbecue Beef the last time, 55% ordered it the next time, 20% switched to the Original, and 25% switched to Italian.. Using an initial matrix for a person who buys an Italian Sandwich on Monday, what are the chances they get Italian again on Wednesday?

a)

28.5%

b)

27.8%

c)

28.8%

d)

27.1%

67.

The Super X sells three kinds of sandwiches – Super X Original, Italian Special, and Barbecue Beef. The Super X clerk observed what people purchased depended on what they had ordered on the previous visit. He conducted a survey and found that of the students who ordered the Original on their last visit, 20% ordered it again the next time, whereas 25% switched to Italian and 55% switched to Barbecue Beef. Of the students who ordered the Italian sandwich the last time, 35% did so again the next time, but 45% switched to the Original and 20% switched to the Barbecue Beef. Of the students who got the Barbecue Beef the last time, 55% ordered it the next time, 20% switched to the Original, and 25% switched to Italian. Given a person who buys an Italian Sandwich on Monday. what are the chances they buy an original in the long run?

a)

26.9%

b)

27.8%

c)

45.3%

d)

33.3%

68.

A regular 6-sided die is to be rolled once. What is the probability that a number less than 3 is rolled?

a)

1/3

b)

1/6

c)

1/2

d)

2/3

69.

A coin is to be tossed twice. p(t, t) = ___

a)

1/4

b)

1/2

c)

2

d)

1/8

70.

There are 3 red marbles, 4 blue marbles, 2 green marbles, and 6 clear marbles in a burlap bag. One marble is randomly pulled from the bag. What is the probability the marble was green?

a)

1/2

b)

1/7

c)

2/15

d)

2/17

71.

A jar contains 2 pink, 6 red, and 4 blue marbles. If you pick one marble without looking, what is the probability that the marble you pick will be red or blue?

a)

5/6

b)

1/2

c)

1/3

d)

1/6

72.

A piggy bank contains 4 quarters, 18 dimes, 10 nickels, and 8 pennies. A coin is chosen at random, not replaced, then another is chosen. Find each probability.

P (penny, then dime).

a)

1/8

b)

9/100

c)

6/65

d)

17/195

73.

Find the probability of each set of independent events.

drawing a black checker from a bag of 6 black checkers and 4 red checkers, replacing it, and drawing another black checker.

a)

2/3

b)

9/25

c)

2/5

d)

3/5

74.

Kaylee has 1 red, 3 blue, 2 brown, 5 black, 1 purple, and 4 pairs of pink flip-flops in her closet. Assuming she always replaces them, what is the probability that she will choose a blue pair on Monday and a pink pair on Tuesday?

a)

3/8

b)

4/75

c)

3/64

d)

15/32

75.

A chorus has 50 girls and 35 boys. If two are chosen at random to sing a duet, what is the probability that both will be boys?

a)

1/6

b)

7/17

c)

3/8

d)

13/25

76.

The Washington family is having a new house built. They are trying to decide what type of material to use for the outside of the house. The materials they can choose are brown brick, red brick, stucco, rock or siding. What is the probability that the Washingtons will choose either brick or stucco?

a)

$\frac{1}{5}$

b)

$\frac{2}{5}$

c)

$\frac{3}{5}$

d)

$\frac{4}{5}$

77.

Thirty-two percent of families residing in a rural, farming town own a goat and 41% of families own a dog. Twenty-four percent of families own both types of animals. If a family from this town is chosen at random, what is the probability that it owns either a goat or a dog?

a)

0.15

b)

0.33

c)

0.49

d)

0.97

78.

A disc jockey has to choose three songs for the last few minutes of his evening show. If there are nine songs that he feels are appropriate for that time slot, then how many ways can he choose and arrange to play three of those nine songs?

a)

84

b)

504

c)

9!

d)

3!

79.

How many 2-digit numbers can you make using the digits 1, 2, 3, & 4 without repeating the digits?

a)

90

b)

100

c)

12

d)

24

80.

Robin has five different pairs of shoes that match with 6 different pairs of socks. How many shoes-and-socks combinations can she make if she selects one pair of shoes and one pair of socks?

a)

1

b)

30

c)

5!

d)

6!

81.

How would you solve?
A book club offers a choice of 8 books from a list of 40. In how many ways can a member make a collection?

a)

Permutation

b)

Combination

c)

Factorial

d)

MPC

82.

How would you solve?
How many ways can you line up your eight trophies that you've won over the years on a shelf in your room?

a)

Permutation

b)

Combination

c)

Factorial

d)

MPC

83.

Sarah plays softball over the summer. If there are 10 players on the team, how many ways can the coach make a batting order that includes all players?

a)

1

b)

3628800

c)

100

d)

1814400

84.

There are 125 juniors at the high school. How many ways can they elect a president, vice president,and secretary?

a)

1906500

b)

317750

c)

375

d)

1953125

85.

Determine the number of 4-letter arrangements that can be made using the word RHOMBUS.

a)

5040

b)

35

c)

28

d)

840

86.

Erik gets to choose from five cake flavors, seven icing flavors, and three topping flavors. If he can only choose one of each flavor, how many cakes are possible?

a)

35

b)

573

c)

21

d)

105

87.

There are 12 kinds of Girl Scout cookies for sale. How many ways can Rachel choose three different kinds to buy?

a)

220

b)

36

c)

1320

d)

100

88.

How many different 10-letter arrangements can be made from the word BASKETBALL?

a)

3628802

b)

453600

c)

604800

d)

26300

89.

What does the n stand for in the binomial probability formula?

a)

Number of trials

b)

Number of Successes

c)

Probability of Successes

d)

Probability of Failures

90.

What does the r stand for in the binomial probability formula?

a)

Number of trials

b)

Number of Successes

c)

Probability of Successes

d)

Probability of Failures

91.

A survey found that 25% of pet owners had their pets bathed professionally rather than do it themselves. If 18 pet owners are randomly selected, find the probability that exactly 5 people have their pets bathed professionally

a)

3.42

b)

1.03

c)

0.072

d)

0.199

92.

An algebra 2 test has 5 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 5 questions, what is the probability that you get at least 1 question correct?

a)

0.2373

b)

0.3955

c)

0.7627

d)

0.6045

93.

When rolling two dice, the probability of rolling doubles is ⅙. Suppose that a game player rolls the dice five times, hoping to roll doubles. What is the probability the player gets doubles less than three times in 5 attempts?

a)

0.161

b)

0.965

c)

0.015

d)

0.997

94.

Which descriptions are true for a binomial distribution?

a)

the probability of success must be the same for each event

b)

the probability for each event must be 0.5

c)

there can be only two possible outcomes

d)

The probabilities must add to 1

e)

each trial must be independent

95.

Mary is a good student. The probability that she studies and passes her test is 3/5. If the probability that she studies is 8/9. What is the probability that she passes given that she studies?

a)

27/40

b)

1.48

c)

.008

d)

.0593

96.

What is the probability that a student does play a sport given they do not play an instrument?

a)

.2

b)

.2222

c)

.10

d)

.5

97.

Consider the following table with information about all of the students taking Statistics at Happy High School.

Find P( Full-time | Male)

a)

7/3

b)

3/10

c)

3/7

d)

4/7

98.

Given the following VENN Diagram answer the following.

P( A | B )

a)

2

b)

.4

c)

.6

d)

.8

99.

P(A∣B)=

a)

P(A⋂B)/P(A)

b)

P(A⋂B)/P(B)

c)

P(A⋃B)/P(A)

d)

P(A⋃B)/P(B)

100.

A new credit card has been issued to 2000 customers. Of these customers, 1500 hold a Visa, 500 hold an AA card, and 40 hold a Visa and AA card. Find the probability that a random chosen customer holds an AA, given they hold a Visa.

a)

.0267

b)

37.5

c)

.02

d)

.3333

101.

The weather forecaster predicted it will be 4/5 chance of rain for Monday and 2/3 chance of rain for Tuesday. What is the probability it will rain both days?

a)

1/2

b)

20/15

c)

8/15

d)

6/8

102.

45% total of the children in a school have a dog, 30% total have a cat, and 18% have a dog and cat. What percent of those who have a cat also have a dog?

a)

60%

b)

40%

c)

24%

d)

1.7%

103.

Given P(A) = .8 and P( B| A) = .3, find P( A and B).

a)

2.6

b)

.12

c)

.24

d)

.38

Discrete Math: Review of Units 1-3

Total questions: 103

Worksheet time: 26hrs 45mins

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