Chapter 9

What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.

For the following exercises, assume that there are \(n\) ways an event \(A\) can happen, \(m\) ways an event \(B\) can happen, and that \(A\) and \(B\) are non- overlapping. Use the Addition Principle of counting to explain how many ways event \(A\) or \(B\) can occur.

What is an arithmetic sequence?

Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

What is a sample space?

What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

For the following exercises, assume that there are \(n\) ways an event \(A\) can happen, \(m\) ways an event \(B\) can happen, and that \(A\) and \(B\) are non- overlapping. Use the Multiplication Principle of counting to explain how many ways event \(A\) and \(B\) can occur.

What is the difference between an arithmetic sequence and an arithmetic series?

How is the common ratio of a geometric sequence found?

How is the common difference of an arithmetic sequence found?

Describe three ways that a sequence can be defined.

What is an experiment?

What is the Binomial Theorem and what is its use?

Answer the following questions. When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?

What is the procedure for determining whether a sequence is geometric?

How do we determine whether a sequence is arithmetic?

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times.

When is it an advantage to use the Binomial Theorem? Explain.

Answer the following questions. Describe how the permutation of \(n\) objects differs from the permutation of choosing \(r\) objects from a set of \(n\) objects. Include how each is calculated.

What is the difference between an arithmetic sequence and a geometric sequence?

What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?

The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}6 \\ 2\end{array}\right)\)

Answer the following questions. What is the term for the arrangement that selects \(r\) objects from a set of \(n\) objects when the order of the \(r\) objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?

What is an annuity?

Describe how exponential functions and geometric sequences are similar. How are they different?

Describe how linear functions and arithmetic sequences are similar. How are they different?

What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}5 \\ 3\end{array}\right)\)

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. Let the set \(A=\\{-5,-3,-1,2,3,4,5,6\\}\). How many ways are there to choose a negative or an even number from \(\mathrm{A}\) ?

For the following exercises, express each description of a sum using summation notation. The sum of terms \(m^{2}+3 m\) from \(m=1\) to \(m=5\)

For the following exercises, find the common ratio for the geometric sequence. \(1,3,9,27,81, \ldots\)

For the following exercises, find the common difference for the arithmetic sequence provided. $$ \\{5,11,17,23,29, \ldots\\} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=2^{n}-2 $$

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}7 \\ 4\end{array}\right)\)

For the following exercises, express each description of a sum using summation notation. The sum from of \(n=0\) to \(n=4\) of \(5 n\)

For the following exercises, find the common ratio for the geometric sequence. \(-0.125,0.25,-0.5,1,-2, \ldots\)

For the following exercises, find the common difference for the arithmetic sequence provided. $$ \left\\{0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\right\\} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-\frac{16}{n+1} $$

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}9 \\ 7\end{array}\right)\)

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many ways are there to pick a red ace or a club from a standard card playing deck?

For the following exercises, express each description of a sum using summation notation. The sum of \(6 k-5\) from \(k=-2\) to \(k=1\)

For the following exercises, find the common ratio for the geometric sequence. \(-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128}, \ldots\)

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference. $$ \\{11.4,9.3,7.2,5.1,3, \ldots\\} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-(-5)^{n-1} $$

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{c}10 \\ 9\end{array}\right)\)

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?

For the following exercises, express each description of a sum using summation notation. The sum that results from adding the number 4 five times

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(-6,-12,-24,-48,-96, \ldots\)