Chapter 11

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n} .\) 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?

Evaluate \(\frac{n !}{(n-r) !}\) for \(n=20\) and \(r=3\)

What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{100}{n} \quad n:[0,1000,100] \text { by } a_{n}:[0,1,0.1] $$

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

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 \(11,33,99,297, \ldots\)

Evaluate \(\frac{n !}{(n-r) ! r !}\) for \(n=8\) and \(r=3\)

What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{2 n^{2}+5 n-7}{n^{3}} \quad n:[0,10,1] \text { by } a_{n}:[0,2,0.2] $$

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?

What is an annuity?

What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1] $$

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

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.

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?

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

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 \(\sum_{i=1}^{8}(i+7)\) is \(92,\) but the value of \(\sum_{i=1}^{8} i+7\) is 43

Solve and determine whether $$8(x-3)+4=8 x-21$$ is an identity, a conditional equation, or an inconsistent equation.

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.

If \(f(x)=4 x^{2}-5 x-2,\) find $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ and simplify.

Expand: \(\log _{7}\left(\frac{\sqrt[5]{x}}{49 y^{10}}\right)\).

Solve: \(\quad \cos 2 x+3 \sin x-2=0,0 \leq x<2 \pi\).

In Exercises \(99-100,\) use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of fand discuss its relationship to the sum of the given series. Function Series $$ f(x)=\frac{2\left[1-\left(\frac{1}{3}\right)^{x}\right]}{1-\frac{1}{3}} \quad 2+2\left(\frac{1}{3}\right)+2\left(\frac{1}{3}\right)^{2}+2\left(\frac{1}{3}\right)^{3}+\cdots $$

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

The figure shows that when a die is rolled, there are six equally likely outcomes: 1, 2, 3, 4, 5, or 6. What fraction of the outcomes is less than 5?

In Exercises \(99-100,\) use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of fand discuss its relationship to the sum of the given series. Function Series $$ f(x)=\frac{4\left[1-(0.6)^{x}\right]}{1-0.6} \quad 4+4(0.6)+4(0.6)^{2}+4(0.6)^{3}+\cdots $$

The figure shows that when a die is rolled, there are six equally likely outcomes: 1, 2, 3, 4, 5, or 6. What fraction of the outcomes is not less than 5?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

Enough curiosities involving the Fibonacci sequence exist to warrant a flourishing Fibonacci Association, which publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the Internet or the research department of your library, and find one property that interests you. After doing this research, get together with your group to share these intriguing properties.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I modeled California's population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.

$$ \text { Solve: } \frac{x}{x-3}=\frac{2 x}{x-3}-\frac{5}{3} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.

$$ \text { Solve: } x^{4}-6 x^{3}+4 x^{2}+15 x+4=0 $$

Determine the amplitude, period, and phase shift of \(y=\frac{1}{2} \cos \left(3 x+\frac{\pi}{2}\right) .\) Then graph one period of the function. (Section 5.5, Example 6 )

will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots . .\) Find \(a_{2}-a_{1}\) \(a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

In Exercises \(105-108\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\)

will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Use the formula \(a_{n}=4+(n-1)(-7)\) to find the eighth term of the sequence \(4,-3,-10, \ldots\)

You are now 25 years old and would like to retire at age 55 with a retirement fund of \(\$ 1,000,000 .\) How much should you deposit at the end of each month for the next 30 years in an IRA paying \(10 \%\) annual interest compounded monthly to achieve your goal? Round up to the nearest dollar.

Find the dimensions of a rectangle whose perimeter is 22 feet and whose area is 24 square feet. (Section \(8.4,\) Example 5 )

Solve using matrices. Use Gaussian elimination with backsubstitution or Gauss- Jordan elimination. $$ \left\\{\begin{aligned} x-2 y+z &=-4 \\ 2 x+2 y-z &=10 \\ 4 x-y+2 z &=-1 \end{aligned}\right. $$ (Section \(9.1,\) Examples 3 and 5 )

Convert the equation $$ 4 x^{2}+y^{2}-24 x+6 y+9=0 $$ to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of the foci. (Section \(10.1,\) Example 5 )

Use a right triangle to write \(\cos \left(\tan ^{-1} x\right)\) as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x . \text { (Section } 5.7, \text { Example } 9)\)

Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$ is true for the given value of \(n\) $$ n=3: \text { Show that } 1+2+3=\frac{3(3+1)}{2} $$

Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$ is true for the given value of \(n\) $$ n=5: \text { Show that } 1+2+3+4+5=\frac{5(5+1)}{2} $$

Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$ is true for the given value of \(n\) $$ \text { Simplify: } \frac{k(k+1)(2 k+1)}{6}+(k+1)^{2} $$