Chapter 8

Use a formula to find the sum of the finite geometric series. $$ 0.6+0.3+0.15+0.075+0.0375 $$

Use Pascal's triangle to help expand the expression. $$ (4 x-3 y)^{4} $$

In 2004 , the death rate per \(100,000\) people between the ages of 20 and 24 was 94 . What is the probability that a person selected at random from this age group died during \(2004 ?\) (Source? Department of Health and Human Services.)

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$-\frac{1}{4},-\frac{1}{2},-1,-2,-4$$

Evaluate the expression. \(P(10,4)\)

Use a formula to find the sum of the finite geometric series. The first 20 terms of the series defined by \(a_{n}=3(2)^{n-1}\)

Use Pascal's triangle to help expand the expression. $$ (3-2 x)^{5} $$

Find the probability of tossing a coin \(n\) times and obtaining \(n\) heads. What happens to this probability as \(n\) increases? Does this agree with your intuition? Explain.

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$9,6,4, \frac{8}{3}, \frac{16}{9}$$

Evaluate the expression. \(P(34,2)\)

Use a formula to find the sum of the finite geometric series. The first 15 terms of the series defined by \(a_{n}=2\left(\frac{1}{3}\right)^{n}\)

Use Pascal's triangle to help expand the expression. $$ (m+n)^{6} $$

\(2005,\) a total of \(984,155\) cases of AIDS had been diagnosed. The table lists AIDS cases diagnosed in certain cities. Estimate the probability that a person diagnosed with AIDS satisfied the following conditions. $$ \begin{array}{|lc|} \hline \text { New York } & 158,502 \\ \hline \text { Los Angeles } & 49,913 \\ \hline \text { San Francise } & 30,277 \\ \hline \text { Miami } & 29,092 \\ \hline \end{array} $$ A. Resided in New York B. Did not reside in New York C. Resided in Los Angeles or Miami

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=5, d=-2$$

How many ways can 4 people stand in a line?

Use a formula to approximate the sum for \(n=4,7, \text { and } 10.\) $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots+\left(-\frac{1}{2}\right)^{n-1} $$

Use Pascal's triangle to help expand the expression. $$ (2 m-n)^{4} $$

Find the probability of rolling a die five times and obtaining a 6 on the first two rolls, a 5 on the third roll, and a \(1,2,3,\) or 4 on the last two rolls.

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=-3, d=5$$

How many arrangements are there of 6 different books on a shelf?

Use Pascal's triangle to help expand the expression. $$ \left(2 x^{3}-y^{2}\right)^{3} $$

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{3}=1, d=3$$

In how many arrangements can 3 students from a class of 15 each give a speech?

Use Pascal's triangle to help expand the expression. $$ \left(3 x^{2}+y^{3}\right)^{4} $$

Two dice are rolled. Find the probability that the dice show a sum other than 7 or 11

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{4}=12, d=-10$$f

How many ways could 5 basketball players be introduced at a game?

Find the specified term. The fourth term of \((a+b)^{9}\)

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{2}=5, a_{6}=13$$

(Refer to the discussion after Example \(4 .\) ) A salesperson must travel to 3 of 7 cities. Direct travel is possible between every pair of cities. How many arrangements are there in which the salesperson could visit these 3 cities? Assume that traveling a route in reverse order constitutes a different arrangement.

Find the sum of the infinite geometric series. $$ 1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\cdots $$

Find the specified term. The second term of \((m-n)^{9}\)

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{3}=22, a_{17}=-20$$

Find the sum of the infinite geometric series. $$ 5+\frac{5}{2}+\frac{5}{4}+\frac{5}{8}+\frac{5}{16}+\cdots $$

Find the specified term. The fifth term of \((x+y)^{8}\)

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=8, a_{4}=17$$

How many 7-digit phone numbers are there if the first three numbers must be \(387,388,\) or \(389 ?\)

Find the sum of the infinite geometric series. $$ 6-4+\frac{8}{3}-\frac{16}{9}+\frac{32}{27}-\frac{64}{81}+\cdots $$

Find the specified term. The third term of \((a+b)^{7}\)

Unfair Coin Suppose a coin is not fair, but instead the probability of obtaining a head (H) is \(\frac{3}{4}\) and a tail ( \(\mathbf{T}\) ) is \(\frac{1}{4}\). What is the probability of each event? A. \(\mathrm{HT}\) B. \(\mathrm{HH}\) C. HHT D. THT

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=-2, a_{5}=8$$

Find the sum of the infinite geometric series. $$ -2+\frac{1}{2}-\frac{1}{8}+\frac{1}{32}-\frac{1}{128}+\cdots $$

Find the specified term. The fourth term of \((2 x+y)^{5}\)

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{5}=-4, a_{8}=-2.5$$

How many ways can 7 people sit at a round table? (For a way to be different, at least one person must be sitting next to someone different.)

Find the sum of the infinite geometric series. $$ 1-\frac{1}{10}+\frac{1}{100}-\frac{1}{1000}+\dots+\left(-\frac{1}{10}\right)^{n-1}+\cdots $$

Find the specified term. The eighth term of \((2 a-b)^{9}\)

Code The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. Each setting represents a different code. What is the probability of guessing someone's code at random? (Source: Promax.)

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{3}=10, a_{7}=-4$$

A softball team has 10 players. How many batting orders are possible?