Chapter 11

Describe how you would use mathematical induction to prove $$ \begin{array}{l} {(a+b)^{n}=\left(\begin{array}{c} {n} \\ {0} \end{array}\right) a^{n}+\left(\begin{array}{c} {n} \\ {1} \end{array}\right) a^{n-1} b+\left(\begin{array}{c} {n} \\ {2} \end{array}\right) a^{n-2} b^{2}} \\ {+\cdots+\left(\begin{array}{c} {n} \\ {n-1} \end{array}\right) a b^{n-1}+\left(\begin{array}{c} {n} \\ {n} \end{array}\right) b^{n}} \end{array} $$ What happens when n = 1? Write the statement that we assume to be true. Write the statement that we must prove. What must be done to the left side of the assumed statement to make it look like the left side of the statement that must be proved? (More detail on the actual proof is found in Exercise 85.)

Baskin-Robbins offers 31 different flavors of ice cream. One of its items is a bowl consisting of three scoops of ice cream, each a different flavor. How many such bowls are possible?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Assuming the next U.S. president will be a Democrat or a Republican, the probability of a Republican president is 0.5

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) 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. 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.

In how many ways can these six jokes be ranked from best to worst?

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) 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?

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$f_{1}(x)=(x+2)^{3} \quad f_{2}(x)=x^{3}$$ $$f_{3}(x)=x^{3}+6 x^{2} \quad f_{4}(x)=x^{3}+6 x^{2}+12 x$$ $$f_{5}(x)=x^{3}+6 x^{2}+12 x+8$$ Use a \([-10,10,1]\) by \([-30,30,10]\) viewing rectangle.

If Phyllis Diller’s joke about books is excluded, in how many ways can the remaining five jokes be ranked from best to worst?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I toss a coin, the probability of getting heads or tails is 1 but the probability of getting heads and tails is \(0 .\)

The table shows the population of California for 2000 and \(2010,\) with estimates given by the U.S. Census Bureau for 2001 through 2009 \(\begin{array}{lllllll}\hline \text { Year } & {2000} & {2001} & {2002} & {2003} & {2004} & {2005} \\ \hline \text { Population } & {33.87} & {34.21} & {34.55} & {34.90} & {35.25} & {35.60} \\ \hline\end{array}\) \(\begin{array}{llllll}{\text { Year }} & {2006} & {2007} & {2008} & {2009} & {2010} \\ {\text { Population }} & {36.00} & {36.36} & {36.72} & {37.09} & {37.25}\end{array}\) 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.

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

You are considering two job offers. Company A will start you at \(\$ 19,000\) a year and guarantee a raise of \(\$ 2600\) per year. Company B will start you at a higher salary, \(\$ 27,000\) a year, but will only guarantee a raise of \(\$ 1200\) per year. Find the total salary that each company will pay over a ten- year period. Which company pays the greater total amount?

The table shows the population of Texas for 2000 and 2010 with estimates given by the U.S. Census Bureau for 2001 through 2009 \(\begin{array}{llllll}{\text { Year }} & {2000} & {2001} & {2002} & {2003} & {2004} \\ \hline \text { Population } & {20.85} & {21.27} & {21.70} & {22.13} & {22.57} & {23.02} \\\ \hline\end{array}\) \(\begin{array}{llllll}{\text { Year }} & {2006} & {2007} & {2008} & {2009} & {2010} \\ \hline \text { Population } & {23.48} & {23.95} & {24.43} & {24.92} & {25.15} \\ {\text { in millions }} & {23.48} & {23.95} & {24.43} & {24.92} & {25.15}\end{array}\) 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.

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 68 and 69 to verify the expansion. $$ f_{1}(x)=(x-1)^{3} $$

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?

Suppose that it is a drawing in which the Powerball jackpot is promised to exceed \(\$ 700\) million. If a person purchases \(292,201,338\) tickets at \(\$ 2\) per ticket (all possible combinations), isn't this a guarantee of winning the jackpot? Because the probability in this situation is \(1,\) what's wrong with doing this?

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 68 and 69 to verify the expansion. $$ f_{1}(x)=(x-2)^{4} $$

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 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.

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

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\) In Exercises \(71-72,\) 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?

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 $$ a_{n}=10,000\left(1+\frac{0.08}{4}\right)^{n}, \quad n=1,2,3, \dots $$ Find the balance in the account after six years. Round to the nearest cent.

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 68 and 69 to verify the expansion. $$ f_{1}(x)=(x+2)^{6} $$

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

In a class of 50 students, 29 are Democrats, 11 are business majors, and 5 of the business majors are Democrats. If one student is randomly selected from the class, find the probability of choosing a. a Democrat who is not a business major. b. a student who is neither a Democrat nor a business major.

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). 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,

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to expand \(\left(x^{3}-y^{4}\right)^{5},\) I find it helpful to rewrite the expression inside the parentheses as \(x^{3}+\left(-y^{4}\right)\)

What is the common difference in an arithmetic sequence?

Explain the Fundamental Counting Principle.

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.

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.

On New Year's Eve, the probability of a person driving while intoxicated or having a driving accident is \(0.35 .\) If the probability of driving while intoxicated is 0.32 and the probability of having a driving accident is \(0.09,\) find the probability of a person having a driving accident while intoxicated.

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). 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 \(\mathrm{B}\) 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.

a. If two people are selected at random, the probability that they do not have the same birthday (day and month) is and \(y\) ad not not mone monn \(365^{3}-\frac{364}{365}\). Explain why this is so. (Ignore leap years and ex 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?

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A pendulum swings through an arc of 20 inches. On each successive swing, the length of the arc is \(90 \%\) of the previous length. $$ \begin{array}{cccc} {20,} & {0.9(20),} & {0.9^{2}(20),} & {0.9^{3}(20)} \\ {\text {1st }} & {2 \text { nd }} & {3 \text { rd }} & {4 \text { th }} \\ {\text { swing }} & {\text { swing }} & {\text { swing }} & {\text { swing }} \end{array} $$ After 10 swings, what is the total length of the distance the pendulum has swung?

What does the graph of a sequence look like? How is it obtained?

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.

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?

After a \(20 \%\) reduction, a digital camera sold for \(\$ 256\) What was the price before the reduction?

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. $$ \begin{array}{cccc} {16,} & {0.96(16),} & {(0.96)^{2}(16),} & {(0.96)^{3}(16)} \\ {\text { Ist }} & {2 \text { nd }} & {3 \text { rd }} & {4 \text { th }} \\ {\text { swing }} & {\text { swing }} & {\text { swing }} & {\text { swing }} \end{array} $$ After 10 swings, what is the total length of the distance the pendulum has swung?

What is a recursion formula?

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

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

Find the average rate of change of \(f(x)=x^{2}-1\) from \(x_{1}=1\) to \(x_{2}=2\)

Use the formula for the value of an annuity to solve Exercises 77–84. 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.

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

Graph \(f(x)=x^{2} .\) Then use the graph of \(f\) to obtain the graph of of \(g(x)=(x+2)^{2}-1\)