Chapter 11

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 \(\$ 2500\) 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.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Rather than performing the addition, I used the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\) to find the sum of the first 30 terms of the sequence \(2,4,8,16,32,\)

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The Binomial Theorem can be written in condensed form as $$ (a+b)^{n}=\sum_{r=0}^{n}\left(\begin{array}{l} {n} \\ {r} \end{array}\right) a^{n-r} b^{r} $$

What is a combination?

Solve triangle \(A B C\) if \(a=17, b=28,\) and \(c=15\). Round angle measures to the nearest degree.

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

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.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To find the fifth term in the expansion of \((2 x+3 y)^{7}\) we use the formula for finding a particular term with \(r=5\) \(a=2 x, b=3 y,\) and \(n=7\)

Explain how to distinguish between permutation and combination problems.

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?

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

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 \(n\) th term of the sequence in terms of the value of your car at the end of each year.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There are no values of \(a\) and \(b\) such that $$ (a+b)^{4}=a^{4}+b^{4} $$

Write a word problem that can be solved by evaluating \(_{7} C_{3}\)

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

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

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

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

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

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

Here are two ways of investing \(\$ 30,000\) for 20 years. $$ \begin{array}{ccc} {\text { Lump-Sum Deposit }} & {\text { Rate }} & {\text { Time }} \\ {\$ 30,000} & {5 \% \text { compounded }} & {20 \text { years }} \\ {} & {\text { annually }} \end{array} $$ $$ \begin{array}{ll} {\text { Periodic Deposits }} & {\text { Rate } \quad \text { Time }} \\ {\$ 1500 \text { at the end }} & {5 \% \text { compounded } 20 \text { years }} \\ {\text { of each year }} & {\text { annually }} \end{array} $$ After 20 years, how much more will you have from the lump-sum investment than from the annuity?

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 \(10 ?\)

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. $$ \frac{20 !}{300} $$

Here are two ways of investing \(\$ 40,000\) for 25 years. \(\begin{array}{cccc}{\text { Lump-Sum Deposit }} & {\text { Rate }} & {\text { Time }} \\ {\$ 40,000} & {6.5 \% \text { compounded }} & {25 \text { years }} \\\ {} & {\text { annually }}\end{array}\) $$ \begin{array}{ll} {\text { Periodic Deposits }} & {\text { Rate } \quad \text { Time }} \\ {\$ 1600 \text { at the end }} & {6.5 \% \text { compounded } 25 \text { years }} \\ {\text { of each year }} & {\text { annually }} \end{array} $$ After 25 years, how much more will you have from the lump-sum investment than from the annuity?

Show that the sum of the first \(n\) positive odd integers, $$ 1+3+5+\dots+(2 n-1) $$ is n2.

Show that $$ \left(\begin{array}{l} {n} \\ {r} \end{array}\right)+\left(\begin{array}{c} {n} \\ {r+1} \end{array}\right)=\left(\begin{array}{c} {n+1} \\ {r+1} \end{array}\right) $$

use a calculator’s factorial key to evaluate each expression. $$ \frac{20 !}{(20-3) !} $$

A new factory in a small town has an annual payroll of S6 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?

Write an equation in point-slope form and slope-intercept form for the line passing through \((-2,-6)\) and perpendicular to the line whose equation is \(x-3 y+9=0 .\) (Section 2.4 Example \(2)\)

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 $$ \begin{array}{l} {(a+b)^{k+1}=\left(\begin{array}{c} {k} \\ {0} \end{array}\right) a^{k+1}+\left[\left(\begin{array}{c} {k} \\ {0} \end{array}\right)+\left(\begin{array}{c} {k} \\ {1} \end{array}\right)\right] a^{k} b} \\ {+\left[\left(\begin{array}{c} {k} \\ {1} \end{array}\right)+\left(\begin{array}{c} {k} \\ {2} \end{array}\right)\right] a^{k-1} b^{2}+\left[\left(\begin{array}{c} {k} \\ {2} \end{array}\right)+\left(\begin{array}{c} {k} \\ {3} \end{array}\right) a^{k-2} b^{3}\right.} \\ {+\cdots+\left[\left(\begin{array}{c} {k} \\ {k-1} \end{array}\right)+\left(\begin{array}{c} {k} \\ {k} \end{array}\right)\right] a b^{k}+\left(\begin{array}{c} {k} \\ {k} \end{array}\right) b k+1} \end{array} $$ Use the result of Exercise 84 to add the binomial sums in brackets. For example, because \(\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)+\left(\begin{array}{c}{n} \\ {r+1}\end{array}\right)\) $$ \begin{aligned} &=\left(\begin{array}{l} {n+1} \\ {r+1} \end{array}\right), \text { then }\left(\begin{array}{l} {k} \\ {0} \end{array}\right)+\left(\begin{array}{l} {k} \\ {1} \end{array}\right)=\left(\begin{array}{c} {k+1} \\ {1} \end{array}\right) \text { and }\\\ &\left(\begin{array}{l} {k} \\ {1} \end{array}\right)+\left(\begin{array}{l} {k} \\ {2} \end{array}\right)=\left(\begin{array}{c} {k+1} \\ {2} \end{array}\right) \end{aligned} $$ f. Because \(\left(\begin{array}{c}{k} \\\ {0}\end{array}\right)=\left(\begin{array}{c}{k+1} \\\ {0}\end{array}\right)(\text { why } ?)\) and \(\left(\begin{array}{l}{k} \\\ {k}\end{array}\right)=\left(\begin{array}{c}{k+1} \\ {k+1}\end{array}\right)\) (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 a calculator’s factorial key to evaluate each expression. $$ \frac{54 !}{(54-3) ! 3 !} $$

I used the combinations formula to determine how many different four-note sound sequences can be created from the notes C, D, E, F, G, A, and B.

Among all pairs of numbers whose sum is \(24,\) find a pair whose product is as large as possible. What is the maximum product? (Section 3.1, Example 6)

Solve: $$ 6|1-2 x|-7=11 $$

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.

$$ \text { Solve: } \quad \log _{2}(x+9)-\log _{2} x=1 . \text { (Section } 4.4, \text { Example } 7 \text { ) } $$

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

Graph \(y=3 \tan \frac{x}{2}\) for \(-\pi

Graph: \(f(x)=-2(x-1)^{2}(x+3)\)

As \(n\) increases, the terms of the sequence $$ a_{n}=\left(1+\frac{1}{n}\right)^{n} $$ get closer and closer to the number \(e\) (where \(e=2.7183\) ). Use a calculator to find \(a_{10}, a_{100}, a_{1000}, a_{10000}\) and \(a_{10000}\) comparing these terms to your calculator's decimal approximation for \(e .\)

What is the common ratio in a geometric sequence?

Find the exact value of \(\cos 75^{\circ}\) using \(\cos 75^{\circ}=\cos \left(120^{\circ}-45^{\circ}\right)\) and the difference formula for cosines. (Section 6.2 Example 1

What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{n}{n+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1] $$

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