Chapter 8

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 50\) at the end of each month in an IRA that pays \(5.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) 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. The sequence for the number of seats per row in our movie theater as the rows move toward the back is arithmetic with \(d-1\) so people don't block the view of those in the row behind them.

You buy a new car for \(\$ 24,000 .\) At the end of \(n\) years, the value of your car is given by the sequence $$a_{n}-24,000\left(\frac{3}{4}\right)^{n}, \quad n-1,2,3, \dots$$ Find \(a_{5}\) and write a sentence explaining what this value represents. Describe the \(n\) the term of the sequence in terms of the value of your car at the end of each year.

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 75\) at the end of each month in an IRA that pays \(6.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) 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. Beginning at 6: 45 A.M.., a bus stops on my block every 23 minutes, so 1 used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.

Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\) Hint: Write \(x^{2}+x+1\) as \(x^{2}+(x+1)\).

Use a calculator's factorial key to evaluate each expression. $$\frac{200 !}{198 !}$$

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. To offer scholarship funds to children of employees, a company invests \(\$ 10,000\) at the end of every three months in an annuity that pays \(10.5 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.

In the sequence \(21,700,23,172,24,644,26,116, \dots,\) which term is \(314,628 ?\)

Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.

Use a calculator's factorial key to evaluate each expression. $$\left(\frac{300}{20}\right)!$$

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. To offer scholarship funds to children of employees, a company invests \(\$ 15,000\) at the end of every three months in an annuity that pays \(9 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.

A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January 10 , how many degree-days are included from January 1 to January \(10 ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the Fundamental Counting Principle to determine the number of five- digit ZIP codes that are available to the U.S. Postal Service.

Prove that $$\left(\begin{array}{l}n \\\r\end{array}\right)-\left(\begin{array}{c}n \\\n-r\end{array}\right)$$.

Use a calculator's factorial key to evaluate each expression. $$\frac{20 !}{300}$$

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. Here are two ways of investing \(\$ 30,000\) for 20 years. $$ \begin{array}{ccc} \hline \text { Lump-Sum Deposit } & \text { Rate } & \text { Time } \\ \$ 30,000 & 5 \% \text { compounded } & 20 \text { years } \\ & \text { annually } & \end{array} $$ $$ \begin{array}{ccc} \hline \text { Periodic Deposit } & \text { Rate } & \text { Time } \\ \$ 1500 \text{at the end of the year} & 5 \% \text { compounded } & 20 \text { years } \\ & \text { annually } & \end{array} $$ After 20 years, how much more will you have from the lump-sum investment than from the annuity?

Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\dots+(2 n-1)$$ is \(n^{2}\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the permutations formula to determine the number of ways the manager of a baseball team can form a 9 -player batting order from a team of 25 players.

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. Here are two ways of investing \(\$ 40,000\) for 25 years. $$ \begin{array}{ccc} \hline \text { Lump-Sum Deposit } & \text { Rate } & \text { Time } \\ \$ 40,000 & 6.5 \% \text { compounded } & 25 \text { years } \\ & \text { annually } & \end{array} $$ $$ \begin{array}{lll} \hline \text { Periodic Deposits } & \text { Rate } & \text { Time } \\ \hline \$ 1600 \text { at the end } & 6.5 \% \text { compounded } & 25 \text { years } \\ \text { of each year } & \text { annually } & \end{array} $$ After 25 years, how much more will you have from the lump-sum investment than from the annuity?

Use a calculator's factorial key to evaluate each expression. $$\frac{20 !}{(20-3) !}$$ (GRAPH CANNOT COPY)

Exercises \(85-87\) will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots .\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}}\)and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the combinations formula to determine how many different four-note sound sequences can be created from the notes \(\mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}, \mathrm{A},\) and \(\mathrm{B}\)

Follow the outline below and use mathematical induction to prove the Binomial Theorem: $$\begin{aligned}(a+b)^{n} &-\left(\begin{array}{c}n \\\0\end{array}\right) a^{n}+\left(\begin{array}{c}n \\\1\end{array}\right) a^{n-1} b+\left(\begin{array}{c}n \\\2\end{array}\right) a^{n-2} b^{2} \\\&+\cdots+\left(\begin{array}{c}n \\\n-1\end{array}\right) a b^{n-1}+\left(\begin{array}{c}n \\\n\end{array}\right) b^{n}\end{aligned}$$ a. Verify the formula for \(n-1\) b. Replace \(n\) with \(k\) and write the statement that is assumed true. Replace \(n\) with \(k+1\) and write the statement that must be proved. c. Multiply both sides of the statement assumed to be true by \(a+b .\) Add exponents on the left. On the right, distribute \(a\) and \(b,\) respectively. d. Collect like terms on the right. At this point, you should have $$\begin{array}{l}(a+b)^{k+1}-\left(\begin{array}{c}k \\\0\end{array}\right)a^{k+1}+\left[\left(\begin{array}{c}k \\\0\end{array}\right)+\left(\begin{array}{c}k \\\1\end{array}\right)\right] a^{k} b \\\\+\left[\left(\begin{array}{c}k \\\1\end{array}\right)+\left(\begin{array}{c}k \\\2\end{array}\right)\right] a^{k-1} b^{2}+\left[\left(\begin{array}{c}k \\\2\end{array}\right)+\left(\begin{array}{c}k \\\3\end{array}\right)\right] a^{k-2} b^{3} \\\\+\cdots+\left[\left(\begin{array}{c}k \\\k-1\end{array}\right)+\left(\begin{array}{c}k \\\k\end{array}\right)\right] a b^{k}+\left(\begin{array}{c}k \\\k\end{array}\right) b^{k+1}\end{array}$$ e. Use the result of Exercise 84 to add the binomial sums in brackets. For example, because \(\left(\begin{array}{l}n \\\ r\end{array}\right)+\left(\begin{array}{c}n \\ r+1\end{array}\right)\) $$\begin{aligned}&-\left(\begin{array}{l}n+1 \\\r+1\end{array}\right), \text { then }\left(\begin{array}{l}k \\\0\end{array}\right)+\left(\begin{array}{l}k \\\1\end{array}\right)-\left(\begin{array}{c}k+1 \\\1\end{array}\right) \text { and }\\\&\left(\begin{array}{l}k \\\1\end{array}\right)+\left(\begin{array}{l}k \\\2\end{array}\right)-\left(\begin{array}{c}k+1 \\\2\end{array}\right)\end{aligned}$$ f. Because \(\left(\begin{array}{l}k \\\ 0\end{array}\right)-\left(\begin{array}{c}k+1 \\ 0\end{array}\right)(\text { why? })\) and \(\left(\begin{array}{l}k \\\ k\end{array}\right)-\left(\begin{array}{l}k+1 \\ k+1\end{array}\right)\) (why?), substitute these results and the results from part (e) into the equation in part (d). This should give the statement that we were required to prove in the second step of the mathematical induction process.

Use the formula for the sum of an infinite geometric series to solve. A new factory in a small town has an annual payroll of \(\$ 6\) million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?

Use a calculator's factorial key to evaluate each expression. $$\frac{54 !}{(54-3) ! 3 !}$$ (GRAPH CANNOT COPY)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the permutations formula to determine the number of ways people can select their 9 favorite baseball players from a team of 25 players.

Exercises 86-88 will help you prepare for the material covered in the next section. $$\text { Evaluate } \frac{n !}{(n-r) !} \text { for } n-20 \text { and } r-3$$.

Exercises \(85-87\) will help you prepare for the material covered in the next section. Use the formula \(a_{n}-a_{1} 3^{n-1}\) to find the seventh term of the sequence \(11,33,99,297, \dots\)

What is a geometric sequence? Give an example with your explanation.

What is the common ratio in a geometric sequence?

Explain how to find the general term of a geometric sequence.

Five men and five women line up at a checkout counter in a store. In how many ways can they line up if the first person in line is a woman and the people in line alternate woman, man, woman, man, and so on?

Many graphing utilities have a sequence-graphing mode that plots. the terms of a sequence as points on a rectangular coordinate system. Consult your manual, if your graphing utility has this capability, use it to graph each of the sequences. What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}-\frac{2 n^{2}+5 n-7}{n^{3}} n:[0,10,1] \text { by } a_{n}:[0,2,0.2]$$

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

What is an annuity?

A mathematics exam consists of 10 multiple-choice questions and 5 open-ended problems in which all work must be shown. If an examinee must answer 8 of the multiple-choice questions and 3 of the open-ended problems, in how many ways can the questions and problems be chosen?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Now that I've studied sequences, I realize that the joke in this cartoon is based on the fact that you can't have a negative number of sheep.

What is the difference between a geometric sequence and an infinite geometric series?

The group should select real-world situations where the Fundamental Counting Principle can be applied. These could involve the number of possible student ID numbers on your campus, the number of possible phone numbers in your community, the number of meal options at a local restaurant, the number of ways a person in the group can select outfits for class, the number of ways a condominium can be purchased in a nearby community, and so on. Once situations have been selected, group members should determine in how many ways each part of the task can be done. Group members will need to obtain menus, find out about telephone-digit requirements in the community, count shirts, pants, shoes in closets, visit condominium sales offices, and so on. Once the group reassembles, apply the Fundamental Counting Principle to determine the number of available options in each situation. Because these numbers may be quite large, use a calculator.

How do you determine if an infinite geometric series has a sum? Explain how to find the sum of such an infinite geometric series.

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is less than \(5 ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It makes a difference whether or not I use parentheses around the expression following the summation symbol, because the value of \(\sum_{i=1}^{5}(i+7)\) is \(92,\) but the value of \(\sum_{i=1}^{8} i+7\) is $43 .

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is not less than \(5 ?\)

For the first 30 days of a flu outbreak, the number of students on your campus who become ill is increasing. Which is worse: The number of students with the flu is increasing arithmetically or is increasing geometrically? Explain your answer.

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is even or greater than \(3 ?\)

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{n !}{(n-1) !}-\frac{1}{n-1}$$

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{-1}^{2}(-1) / 2-0$$

Use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of \(f\) and discuss its relationship to the sum of the given series. $$ f(x)=\frac{2\left[1-\left(\frac{1}{3}\right)^{x}\right]}{1-\frac{1}{3}} \quad 2+2\left(\frac{1}{3}\right)+2\left(\frac{1}{3}\right)^{2}+2\left(\frac{1}{3}\right)^{3}+\cdots $$

Use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of \(f\) and discuss its relationship to the sum of the given series. $$f(x)-\frac{4\left[1-(0.6)^{x}\right]}{1-0.6} \quad 4+4(0.6)+4(0.6)^{2}+4(0.6)^{3}+\cdots$$