Chapter 8

A professional baseball player signs a contract with a beginning salary of \(\$ 3,000,000\) for the first year and an annual increase of \(4 \%\) per year beginning in the second year. That is, beginning in year \(2,\) the athlete's salary will be 1.04 times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.

A company offers a starting yearly salary of \(\$ 33,000\) with raises of \(\$ 2500\) per year. Find the total salary over a ten-year period.

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller If Phyllis Diller's joke about books is excluded, in how many ways can the remaining five jokes be ranked from best to worst?

In Exercises 68-69, graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. \(f_{1}(x)-(x+2)^{3} \quad f_{2}(x)-x^{3}\) \(f_{3}(x)-x^{3}+6 x^{2} \quad f_{4}(x)-x^{3}+6 x^{2}+12 x\) \(f_{s}(x)-x^{3}+6 x^{2}+12 x+8\) Use a \([-10,10,1]\) by \([-30,30,10]\) viewing rectangle.

Use the formula for the general term (the nth term) of a geometric sequence to solve suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. You are offered a job that pays \(\$ 30,000\) for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year \(2,\) your salary will be 1.05 times what it was in the previous year. What can you expect to earn in your sixth year on the job?

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller In how many ways can people select their three favorite jokes from these comments about books?

Suppose that it is a week in which the cash prize in Florida's LOTTO is promised to exceed \(\$ 50\) million. If a person purchases \(22,957,480\) tickets in LOTTO at \(\$ 1\) per ticket (all possible combinations), isn't this a guarantee of winning the lottery? Because the probability in this situation is 1, what's wrong with doing this?

In Exercises 68-69, graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. \(f_{1}(x)-(x+1)^{4}\) \(f(x)=x^{4}\) \(f_{3}(x)-x^{4}+4 x^{3} \quad f_{4}(x)-x^{4}+4 x^{3}+6 x^{2}\) \(f_{s}(x)-x^{4}+4 x^{3}+6 x^{2}+4 x\) \(f_{\phi}(x)-x^{4}+4 x^{3}+6 x^{2}+4 x+1\) Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.

You will develop geometric sequences that model the population growth for California and Texas, the two most-populated U.S. states. The table shows the population of California for 2000 and \(2010,\) with estimates given by the U.S. Census Bureau for 2001 through 2009 $$ \begin{array}{lllllll} \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \end{array} $$ $$ \begin{array}{llllll} \hline \text { Year } & 2006 & 2007 & 2008 & 2009 & 2010 \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after 1999 c. Use your model from part (b) to project California's population, in millions, for the year \(2020 .\) Round to two decimal places.

A theater has 30 seats in the first row, 32 seats in the second row, increasing by 2 seats per row for a total of 26 rows. How many seats are there in the theater?

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller In how many ways can people select their two favorite jokes from these comments about books?

Some three-digit numbers, such as 101 and \(313,\) read the same forward and backward. If you select a number from all three digit numbers, find the probability that it will read the same forward and backward.

You will develop geometric sequences that model the population growth for California and Texas, the two most-populated U.S. states. The table shows the population of Texas for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through 2009 $$ \begin{array}{llllll} \hline \text { Year } & 2006 & 2007 & 2008 & 2009 & 2010 \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 23.48 & 23.95 & 24.43 & 24.92 & 25.15 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after 1999. c. Use your model from part (b) to project Texas's population, in millions, for the year \(2020 .\) Round to two decimal places.

A section in a stadium has 20 seats in the first row, 23 seats in the second row, increasing by 3 seats each row for a total of 38 rows. How many seats are in this section of the stadium?A section in a stadium has 20 seats in the first row, 23 seats in the second row, increasing by 3 seats each row for a total of 38 rows. How many seats are in this section of the stadium?

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller If the order in which these jokes are told makes a difference in terms of how they are received, how many ways can they be delivered if George Carlin's joke is delivered first and Jerry Seinfeld's joke is told last?

A deposit of \(\$ 6000\) is made in an account that earns \(6 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}-6000\left(1+\frac{0.06}{4}\right)^{n}, \quad n-1,2,3, \ldots$$ Find the balance in the account after five years. Round to the nearest cent.

Use the formula for the sum of the first n terms of a geometric sequence to solve you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, continuing to double your savings each day. What will your total savings be for the first 15 days?

What is an arithmetic sequence? Give an example with your explanation.

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller If the order in which these jokes are told makes a difference in terms of how they are received, how many ways can they be delivered if a joke by a man is told first?

A deposit of \(\$ 10,000\) is made in an account that earns \(8 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}-10,000\left(1+\frac{0.08}{4}\right)^{0}, \quad n-1,2,3, \ldots$$ Find the balance in the account after six years. Round to the nearest cent.

Use the formula for the sum of the first n terms of a geometric sequence to solve you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, continuing to double your savings each day. What will your total savings be for the first 30 days?

What is the common difference in an arithmetic sequence?

Explain the Fundamental Counting Principle.

a. If two people are selected at random, the probability that they do not have the same birthday (day and month) is \(\frac{365}{365} \cdot \frac{364}{565} .\) Explain why this is so. (Ignore leap years and assume 365 days in a year.) b. If three people are selected at random, find the probability that they all have different birthdays. c. If three people are selected at random, find the probability that at least two of them have the same birthday. d. If 20 people are selected at random, find the probability that at least 2 of them have the same birthday. e. How large a group is needed to give a 0.5 chance of at least two people having the same birthday?

In Exercises 73-76, determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to expand \(\left(x^{3}-y^{4}\right)^{3}\). I find it helpful to rewrite the expression inside the parentheses as \(x^{3}+\left(-y^{4}\right)\).

What is a sequence? Give an example with your description.

A job pays a salary of \(\$ 24,000\) the first year. During the next 19 years, the salary increases by \(5 \%\) each year. What is the total lifetime salary over the 20 -year period? Round to the nearest dollar.

Explain how to find the general term of an arithmetic sequence.

Write an original problem that can be solved using the Fundamental Counting Principle. Then solve the problem.

Research and present a group report on state lotteries. Include answers to some or all of the following questions: Which states do not have lotteries? Why not? How much is spent per capita on lotteries? What are some of the lottery games? What is the probability of winning top prize in these games? What income groups spend the greatest amount of money on lotteries? If your state has a lottery, what does it do with the money it makes? Is the way the money is spent what was promised when the lottery first began?

In Exercises 73-76, determine whether each statement makes sense or does not make sense, and explain your reasoning. Without writing the expansion of \((x-1)^{6}\). I can see that the terms have alternating positive and negative signs.

Explain how to write terms of a sequence if the formula for the general term is given.

You are investigating two employment opportunities. Company A offers \(\$ 30,000\) the first year. During the next four years, the salary is guaranteed to increase by \(6 \%\) per year. Company B offers \(\$ 32,000\) the first year, with guaranteed annual increases of \(3 \%\) per year after that. Which company offers the better total salary for a five-year contract? By how much? Round to the nearest dollar.

Explain how to find the sum of the first \(n\) terms of an arithmetic sequence without having to add up all the terms.

What is a permutation?

In Exercises 73-76, determine whether each statement makes sense or does not make sense, and explain your reasoning. I use binomial coefficients to expand \((a+b)^{n}\), where \(\left(\begin{array}{l}n \\ 1\end{array}\right)\) is the coefficient of the first term, \(\left(\begin{array}{l}n \\ 2\end{array}\right)\) is the coefficient of the second term, and so on.

A pendulum swings through an arc of 20 inches. On each successive swing, the length of the arc is \(90 \%\) of the previous length. After 10 swings, what is the total length of the distance the pendulum has swung?

Describe what \(_{n} P_{r}\) represents.

In Exercises 73-76, determine whether each statement makes sense or does not make sense, and explain your reasoning. One of the terms in my binomial expansion is \(\left(\begin{array}{l}7 \\\ 5\end{array}\right) x^{2} y^{4}\).

What is a recursion formula?

A pendulum swings through an are of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. After 10 swings, what is the total length of the distance the pendulum has swung?

Write a word problem that can be solved by evaluating \(_7 P_{3}\)

Explain how to find \(n !\) if \(n\) is a positive integer.

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. To save money for a sabbatical to earn a master's degree, you deposit \(\$ 2000\) at the end of each year in an annuity that pays \(7.5 \%\) compounded annually. a. How much will you have saved at the end of five years? b. Find the interest.

What is a combination?

Explain the best way to evaluate \(\frac{900 !}{899 !}\) without a calculator.

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. To save money for a sabbatical to earn a master's degree, you deposit \(\$ 2500\) at the end of each year in an annuity that pays \(6.25 \%\) compounded annually. a. How much will you have saved at the end of five years? b. Find the interest.

Make Sense? In Exercises \(78-81,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. I was able to find the sum of the first 50 terms of an arithmetic sequence even though I did not identify every term.

Explain how to distinguish between permutation and combination problems.

What is the meaning of the symbol \(\Sigma ?\) Give an example with your description.