Chapter 12

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, n-r)=C(n, r)$$

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence comerges or diverges. If you think it converges, make a conjecture about the mumber to which it converges. $$a_{n}=n(n+2)$$

Five years ago, the population of a city was \(49,000 .\) Each year, the zoning commission permits an increase of 580 in the population. What will the maximum population be 5 years from now?

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence comerges or diverges. If you think it converges, make a conjecture about the mumber to which it converges. $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$

A slide with a uniform slope is to be built on a level piece of land. There are to be 20 equally spaced supports, with the longest support 15 meters long and the shortest 2 meters long. Find the total length of all the supports.

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence comerges or diverges. If you think it converges, make a conjecture about the mumber to which it converges. $$a_{n}=5-\frac{1}{n}$$

Chemical Mixture \(A\) scientist has a vat containing 100 liters of a pure chemical. Twenty liters are drained and replaced with water. After complete mixing, 20 liters of the mixture are again drained and replaced with water. What will be the strength of the mixture after nine such drainings?

Determine the positive integer values of \(n\), where the sequence \(a_{n}\) satisfies the inequality. $$a_{n}<10, \text { where } a_{n}=n+1$$

The average family in the United States spends 150 dollars on food per week. Write a general term \(a_{n}\) for a sequence that gives the spending on food after \(n\) weeks. Find \(a_{4}\) and interpret the result.

Half-Life of a Radioactive Substance The half-life of a radioactive substance is the time it takes for half the substance to decay. Suppose the half-life of a substance is 3 years and \(10^{15}\) molecules of the substance are present initially. How many molecules will be present after 15 years?

Determine the positive integer values of \(n\), where the sequence \(a_{n}\) satisfies the inequality. $$a_{n}<11, \text { where } a_{n}=n^{2}$$

The normal growth pattern for children aged \(3-11\) follows an arithmetic sequence. An increase in height of about 6 centimeters per year is expected. Thus, 6 would be the common difference of the sequence. A child who measures 96 centimeters at age 3 would have his expected height in subsequent years represented by the sequence \(102,108\) \(114,120,126,132,138,144\) Each term differs from the adjacent terms by the common difference, 6. (a) If a child measures 98.2 centimeters at age 3 and 109.8 centimeters at age \(5,\) what would be the common difference of the arithmetic sequence describing his yearly height? (b) What would we expect his height to be at age \(8 ?\)

Depreciation in Value Each year a machine loses \(20 \%\) of the value it had at the beginning of the year. Find the value of the machine at the end of 6 years if it cost \(\$ 100,000\) new.

Determine the positive integer values of \(n\), where the sequence \(a_{n}\) satisfies the inequality. $$a_{n} \geq \frac{1}{5}, \text { where } a_{n}=\frac{n}{1+n^{2}}$$

Find all arithmetic sequences \(a_{1}, a_{2}\) \(a_{3}, \ldots\) such that \(a_{1}^{2}, a_{2}^{2}, a_{3}^{2}, \ldots\) is also an arithmetic sequence.

Determine the positive integer values of \(n\), where the sequence \(a_{n}\) satisfies the inequality. $$a_{n} \geq \frac{9}{10}, \text { where } a_{n}=\frac{n-2}{n+1}$$

Explain why the sequence \(\log 2, \log 4\) \(\log 8, \log 16, \ldots\) is an arithmetic sequence.

MODELING Fruit Fly Populations A population of fruit flies is growing in such a way that each generation is 1.5 times as large as the last generation. If there were 100 flies in the first generation, write a formula for a geometric sequence that gives the population of the \(n\) th generation. Find the population of the fourth generation.

Suppose that \(a_{1}, a_{2}, a_{3}, \ldots\) and \(b_{1}, b_{2}, b_{3}, \ldots\) are arithmetic sequences. Let \(d_{n}=a_{n}+c \cdot b_{n}\), for any real number \(c\) and every positive integer \(n\). Show that \(d_{1}, d_{2}, d_{3}, \ldots\) is an arithmetic sequence.

Estimating Powers of \(e\) The series $$ e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\dots+\frac{a^{n}}{n !} $$ where $$ n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n $$ can be used to estimate the value of \(e^{a}\) for any real number a. Use the first eight terms of this series to approximate each expression. Compare this estimate with the actual value. Give values to six decimal places. (a) \(e\) (b) \(e^{-1}\) (c) \(\sqrt{e}\)

Swing of a Pendulum \(\quad\) A pendulum bob swings through an arc 40 centimeters long on its first swing. Each swing thereafter, it swings only \(80 \%\) as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Suppose an insect population density per acre during year \(n\) can be modeled by the following recursively defined sequence. $$\begin{aligned} &a_{1}=500\\\ &a_{n}=0.8 a_{n-1} \end{aligned}$$ (a) Find the population for \(n=1,2,3\) (b) Graph the sequence for \(n=1,2,3, \ldots, 20 .\) Use the window \([0,21]\) by \([0,550] .\) Interpret the graph.

Height of a Dropped Ball Someone drops a ball from a height of 10 meters and notices that on each bounce the ball returns to about \(\frac{3}{4}\) of its previous height. About how far will the ball travel before it comes to rest? (Hint: Consider the sum of two sequences.) (IMAGE CANT COPY)

Suppose an insect population density in thousands per acre during year \(n\) can be modeled by the following recursively defined sequence. $$\begin{aligned} &a_{1}=8\\\ &a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \text { for } n>1 \end{aligned}$$ (a) Find the population for \(n=1,2,3\) (b) Graph the sequence for \(n=1,2,3, \ldots, 20 .\) Use the window \([0,21]\) by \([0,14] .\) Interpret the graph.

Number of Ancestors Each person has two parents, four grandparents, eight great-grandparents, and so on. What is the total number of ancestors a person has, going back five generations? ten generations? (IMAGE CANT COPY)

If certain bacteria are cultured in a medium with sufficient nutrients, they will double in size and then divide every 40 minutes. Let \(N_{1}\) be the initial number of bacteria cells, \(N_{2}\) the number after 40 minutes, \(N_{3}\) the number after 80 minutes, and \(N_{j}\) the number after \(40(j-1)\) minutes. (a) Write \(N_{j+1}\) in terms of \(N_{j}\) for \(j \geq 1\) (b) Determine the number of bacteria after two hours if \(N_{1}=230\) (c) Graph the sequence \(N_{j}\) for \(j=1,2,3, \ldots, 7 .\) Use the window \([0,10]\) by \([0,15,000]\) (d) Describe the growth of these bacteria when there are unlimited nutrients.

Number of Ancestors Suppose a genealogical website allows you to identify all your ancestors who lived during the past 300 years. Assuming that each generation spans about 25 years, guess the number of ancestors that would be found during the 12 generations. Then use the formula for a geometric series to find the actual value.

Salaries You are offered a 6 -week summer job and are asked to select one of the following salary options. Option \(I: \$ 5000\) for the first day, with a \(\$ 10,000\) raise each day for the remaining 29 days (that is, \(\$ 15,000\) for day \(2, \$ 25,000 \text { for day } 3, \text { and so on })\) Option \(2: 1 \notin\) for the first day, with the pay doubled each day thereafter (that is, \(2 \notin\) for day \(2,4 e\) for day 3 and so on \()\) Which option would you choose?

MODELING Drug Dosage Certain medical conditions are treated with a fixed dose of a drug administered at regular intervals. Suppose that a person is given 2 milligrams of a drug each day and that during each 24 -hour period the body utilizes \(40 \%\) of the amount of drug that was present at the beginning of the period. (a) Show that the amount of the drug present in the body at the end of \(n\) days is $$ \sum_{i=1}^{n} 2(0.6)^{i} $$ (b) What will be the approximate quantity of the drug in the body at the end of each day after the treatment has been administered over a long period?

CONCEPT CHECK Suppose that \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, .\) is a geometric sequence. Is the sequence \(a_{1}, a_{3}, a_{5}, \dots\) geometric?

CONCEPT CHECK Is the sequence \(\log 6, \log 36, \log 1296\) \(\log 1,679,616, \dots\) a geometric sequence?