Deck 9: Discrete Mathematics

Question 1

Prove the following statement by mathematical induction:

Answer 1

Question 2

Determine the sixth partial sum of the geometric sequence

A)
B)
C)
D) 6.48
E) 6.67

Answer 2

6.67

Question 3

Write the statements P₁, P_k , and P_k+1 for the equation as if you were writing a proof by mathematical induction. Do not write a proof.

Answer 3

Question 4

Find the coefficient of the term in the expansion of

A) 3
B) 4
C)
D) 35
E) 210

Question 5

Find the sum of the first 12 terms of the arithmetic sequence:
28,22,16,10, ....

Question 6

Find the sum of the infinite series

Question 7

Assume that the probability that a newborn child is a female is 50% . In a family of four children, what is the probability that
(a) all the children are girls?
(b) at least two of the children are boys?

Question 8

The third and fifth terms of a geometric sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

Question 9

The sum of the squares of the first n positive odd integers is
Use this expression to find the sum

A) 5628
B) 73,150
C) 1425
D) 7,700,625
E) 5776

Question 10

A fair coin is tossed five times. Find the probability of tossing exactly two tails in those five tosses.

Question 11

Question 12

Do the following sequences converge or diverge? If they converge, state the limit.
(a)
(b)
(c)

Question 13

Suppose there is a 40% chance of rain tomorrow. If it rains, there is a 20% chance that all of the rides at an amusement park will be operating. If it doesn't rain, there is a 90% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?

Question 14

The third and fifth terms of an arithmetic sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

Question 15

A computer randomly selects a number from 1 through 50 . Two selections are made by the computer to create an ordered pair of numbers. How many elements are in the sample space for this experiment?

A) 2,500
B) 2,450
C) 2,000
D) 100
E) 50

Question 16

A computer password is 6 entries long. If the password must have 3 letters followed by 3 numbers with repetition of numbers and/or letters permitted, how many different passwords are possible?

Question 17

For the experiment in Problem 3, what is the probability that the computer will not select the same number in both selections?

Question 18

Find the fifteenth term of where is labelled the zeroth term.

Question 19

Expand

Question 20

There are 85 students in a college mathematics class. Five students are chosen to work on a project together. How many different groups of five students are there in this class?

Question 21

Sketch a boxplot for the data in Problem 4 .

Question 22

The sequence 2,6,18,54, .... is geometric.
(a) What is the common ratio of the sequence?
(b) What is a recursive rule for the n th term of the sequence?

Question 23

Twenty-four students participated in a basketball contest. Each student took five shots. The number of baskets scored by each student is shown below. Find the mean, median, and mode for the data.

Question 24

A machine fills 16-ounce soda cans with a mean of 16.06 ounces of soda and a standard deviation of 0.03 . Approximately what percentage of the cans will actually contain less than the advertised less than the advertised 16 ounces of soda?

A) 2.5%
B) 5%
C) 10%
D) 12.5%
E) 15%

Question 25

A hoagie consists of a roll, one type of meat, one type of cheese, and three different toppings. If there are 5 kinds of rolls, 5 kinds of meat, 4 kinds of cheese, and 9 toppings from which to choose, how many different hoagies can be built?

Question 26

Find the standard deviation and variance of the scores on a final exam.
65,88,83,80,78,90,92,85,74,70,95,62

Question 27

The table below shows the tuition fee for one semester at a certain university.

Complete a line graph that shows this university's tuition fee trend.

B. Calculate r . Does the data show strong positive correlation?

Question 28

Use the formulas and
to find the sum in terms
of

Question 29

If the scores on a certain exam are normally distributed with a mean of 79 and a standard deviation of 7 , then 95% of the scores will fall in which interval?

A) [72,86]
B) [58,100]
C) [68.5,89.5]
D) [65,93]
E) [75,83]

Question 30

What is the probability of obtaining exactly four heads
in eleven tosses of a fair coin?

A)
B)
C)
D)
E)

Question 31

A student's grade in a particular class is determined as follows:
mean of three class exams -45%
final exam -25%
project -30%
If a student scored 85,88 , and 76 on the three exams, 90 on the final exam, and 86 on the project, what is the student's grade in the class? (Round your answer to the nearest whole number.)

Question 32

Make a histogram of the data, using interval widths of 10 points.

Question 33

Which one of the following sequences is divergent?

A)
B)
C)
D)
E)

Question 34

Using the binomial theorem, expand the binomial

Question 35

Question 36

Write the series in summation notation and find the sum, assuming the suggested pattern continues.

Question 37

Does correlation imply causation?

Question 38

How many different groups of officers - president, vice-president, treasurer, and secretary - can be formed from an organization with 35 members?

Question 39

Find the five-number summary and the range for the data in the stemplot below.

Question 40

Determine whether the infinite geometric series converges. If it does converge, what is the sum?

Question 41

Find the mean, median, and variance. If necessary, round to the nearest hundredth.

Question 42

Find the probability of obtaining exactly six heads in ten tosses of a fair coin.

A)
B)
C)
D)
E)

Question 43

How many different groups of officers - president, vice-president, treasurer, and secretary - can be formed from an organization with 45 members?

Question 44

Which one of the following sequences is divergent?

A)
B)
C)
D)
E)

Question 45

The sequence 25,21,17,13, \ldots is arithmetic.
(a) What is the common difference of the sequence?
(b) What is a recursive rule for the nth term of the sequence?

Question 46

Question 47

Use the formulas and

of n .

Question 48

Is negative association the same as negative correlation?

Question 49

Question 50

Find the mean, median, and variance. If necessary, round to the nearest hundredth.

Question 51

Question 52

Two dice are rolled. Find the probability of rolling a sum greater than 8.

Question 53

Using the binomial theorem, expand the binomial

Question 54

A hoagie consists of a roll, two types of meat, one type of cheese, and four different toppings. If there are 6 kinds of rolls, 7 kinds of meat, 4 kinds of cheese, and 10 toppings from which to choose, how many different hoagies can be built?

Question 55

Two dice are rolled. Find the probability of rolling a sum greater than 6.

Question 56

Question 57

Determine whether the infinite geometric series converges. If it does converge, what is the sum?

Question 58

Construct a boxplot for the data.

Question 59

Write the series in summation notation and find the sum, assuming the suggested pattern continues.

Question 60

Prove the following statement by mathematical induction.

Question 61

Construct a boxplot for the data.

Question 62

Prove the following statement by mathematical induction.