Chapter 9

Fill in the blank. The _________ states that if there are \(m_{1}\) ways for one event to occur and \(m_{2}\) ways for a second event to occur, then there are \(m_{1} \bullet m_{2}\) ways for both events to occur.

The notation used to denote a binomial coefficient is _______ or _______ .

A sequence is called a ______ sequence if the ratios of consecutive terms are the same. This ratio is called the ______ ratio.

Fill in the blank(s). The function values \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots, a_{n}, \ldots\) are called the _____ of a sequence.

The \(n\) th term of an arithmetic sequence has the form __________.

Fill in the blank(s). The set of all possible outcomes of an experiment is called the _____.

When you write out the coefficients for a binomial that is raised to a power, you are _______ a _______ .

The \(n\)th term of a geometric sequence has the form \(a_{n}=\) ______.

Fill in the blank(s). If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, then the sequence is defined _____.

The formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\) can be used to find the sum of the first \(n\) terms of an arithmetic sequence, called the ____________.

List two ways to find binomial coefficients.

The sum of the terms of an infinite geometric sequence is called a _____ .

Fill in the blank(s). For the sum \(\sum_{i=1}^{n} a_{i}, i\) is called the _____ of summation, \(n\) is the _____ of summation, and 1 is the _____ of summation.

How do you know when a sequence is arithmetic?

Fill in the blank(s). If two events from the same sample space have no outcomes in common, then the two events are _____ .

Is the ordering of \(n\) elements called a permutation or a combination of the elements?

In the expression of \((x+y)^{3},\) what is the sum of the powers of the third term?

Can a geometric sequence have a common ratio of \(0 ?\)

Fill in the blank(s). The sum of the terms of a finite or an infinite sequence is called a _____.

Is 4 or 1 the common difference of the arithmetic sequence \(a_{n}=4 n+1 ?\)

Fill in the blank(s). If the occurrence of one event has no effect on the occurrence of a second event, then the events are _____

What do \(n\) and \(r\) represent in the formula \(_{n} P_{r}=\frac{n !}{(n-r) !} ?\)

Find the binomial coefficient. \(_{7} C_{5}\)

For what values of the common ratio \(r\) is it possible to find the sum of an infinite geometric series?

Which describes an infinite sequence? a finite sequence? (a) The domain consists of the first \(n\) positive integers. (b) The domain consists of the set of positive integers.

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$10,12,14,16,18, \ldots$$

Write an inequality that represents the possible values of the probability \(P(E)\) of an event.

Find the binomial coefficient. \(_{8} C_{6}\)

Which formula represents the sum of a finite geometric sequence? an infinite geometric series? (a) \(S=\frac{a_{1}}{1-r},|r|<1\) (b) \(S_{n}=a_{1}\left(\frac{1-r^{n}}{1-r}\right)\)

Write \(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6\) in factorial notation.

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$4,9,14,19,24, \ldots$$

What is the probability of an impossible event?

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$5,15,45,135, \ldots$$

Find the binomial coefficient. \(20^{C}_{15}\)

Write the first five terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=2 n-5$$

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$3, \frac{5}{2}, 2, \frac{3}{2}, 1, \ldots$$

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$3,12,48,192, \ldots$$

Find the binomial coefficient. \(_{19} C_{12}\)

Write the first five terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=4 n-7$$

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$3.7,3.1,2.5,1.9,1.3, \ldots$$

Match the probability formula with the correct probability name. (a) Probability of the union of two events (i) \(P(A \cup B)=P(A)+P(B)\) (b) Probability of mutually exclusive events (ii) \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\) (c) Probability of independent events (iii) \(P(A \text { and } B)=P(A) \cdot P(B)\)

Find the binomial coefficient. \(_{14} C_{1}\)

Write the first five terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=3^{n}$$

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, \ldots$$

Determine the sample space for the experiment. A coin and a six-sided die are tossed.

Determine the number of ways in which a computer can randomly generate one or more such integers, or pairs of integers, from 1 through 15. A pair of integers whose sum is 20

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$4,19,34,49, \dots$$

Find the binomial coefficient. \(_{18} C_{2}\)

Write the first five terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=\left(\frac{1}{2}\right)^{n}$$

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$\frac{1}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}, \frac{16}{3}, \dots$$