Chapter 8

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

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.

Show that $$ \left(\begin{array}{l}n \\\r\end{array}\right)+\left(\begin{array}{c}n \\\r+1\end{array}\right)=\left(\begin{array}{l}n+1 \\\r+1 \end{array}\right) $$ Hints: $$ \begin{aligned}&(n-r) !=(n-r)(n-r-1) !\\\&(r+1) !=(r+1) r !\end{aligned} $$

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.

Follow the outline on the next page to use mathematical induction to prove that $$ \begin{aligned}(a+b)^{n}=\left(\begin{array}{c}n \\\0\end{array}\right) a^{n}+\left(\begin{array}{c}n \\\1\end{array}\right) a^{n-1} b+\left(\begin{array}{c}n \\\2\end{array}\right) a^{n-2} b^{2} \\\\+\cdots+\left(\begin{array}{c}n \\\n-1\end{array}\right) a b^{n-1}+\left(\begin{array}{c}n \\\n\end{array}\right) b^{n}\end{aligned} $$ 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 $$ \begin{aligned}&(a+b)^{k+1}=\left(\begin{array}{l}k \\\0\end{array}\right) a^{k+1}+\left[\left(\begin{array}{l}k \\\0\end{array}\right)+\left(\begin{array}{l}k \\\1\end{array}\right)\right] a^{k} b\\\&\begin{array}{l}+\left[\left(\begin{array}{c}k \\\1\end{array}\right)+\left(\begin{array}{c}k \\\2\end{array}\right)\right] a^{k-1} b^{2}+\left[\left(\begin{array}{c}k \\\2\end{array}\right)+\left(\begin{array}{c}k \\\3\end{array}\right)\right] a^{k-2} b^{3} \\\\+\cdots+\left[\left(\begin{array}{c}k \\\k-1\end{array}\right)+\left(\begin{array}{c}k \\\k\end{array}\right)\right] a b^{k}+\left(\begin{array}{c}k \\\k\end{array}\right) b^{k+1} \end{array}\end{aligned} $$ e. Use the result of Exercise 74 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)$$=\left(\begin{array}{l}n+1 \\ r+1\end{array}\right),\) then \(\left(\begin{array}{l}k \\ 0\end{array}\right)+\left(\begin{array}{l}k \\\ 1\end{array}\right)=\left(\begin{array}{c}k+1 \\\1\end{array}\right)\) and\(\left(\begin{array}{l}k \\ 1\end{array}\right)+\left(\begin{array}{l}k \\\2\end{array}\right)=\left(\begin{array}{c}k+1 \\ 2\end{array}\right)\) f. Because \(\left(\begin{array}{l}k \\\ 0\end{array}\right)=\left(\begin{array}{c}k+1 \\ 0\end{array}\right) \quad\) (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.

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

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

What is the common ratio in a geometric sequence?

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

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

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

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

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

What is an annuity?

Use a calculator's factorial key to evaluate each expression. $$\frac{54 !}{(54-3) ! 3 !}$$

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

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

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

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 \(e \approx 2.7183\) ). Use a calculator to find \(a_{10}, a_{100}, a_{1000}, a_{10}, a_{00},\) and \(a_{100,000}\) comparing these terms to your calculator's decimal approximation for \(e\).

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.

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? $$a_{n}=\frac{n}{n+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$

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? $$a_{n}=\frac{2 n^{2}+5 n-7}{n^{3}} \quad n:[0,10,1] \text { by } a_{n}:[0,2,0.2]$$

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

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? $$a_{n}=\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$

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

Which one of the following is true? a. \(\frac{n !}{(n-1) !}=\frac{1}{n-1}\) b. The Fibonacei sequence \(1,1,2,3,5,8,13,21,34,55,89\) \(144, \ldots\) can be defined recursively using \(a_{0}=1, a_{1}=1\) \(a_{n}=a_{n-2}+a_{n-1},\) where \(n \geq 2\). c. \(\sum_{i=1}^{2}(-1)^{i} 2^{i}=0\) d. \(\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}\)

Which one of the following is true? a. The sequence \(2,6,24,120, \ldots\) is an example of a geometric sequence. b. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}\). c. \(10-5+\frac{5}{2}-\frac{5}{4}+\cdots=\frac{10}{1-\frac{1}{2}}\) d. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1},\) the common ratio is \(\frac{1}{2}\).

Write the first five terms of the sequence whose first term is 9 and whose general term is $$a_{n}=\left\\{\begin{array}{ll} \frac{a_{n-1}}{2} & \text { if } a_{n-1} \text { is even } \\ 3 a_{n-1}+5 & \text { if } a_{n-1} \text { is odd } \end{array}\right.$$

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.

You are now 25 years old and would like to retire at age 55 with a retirement fund of 1,000,000 dollar. 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 to the nearest dollar.