Chapter 8

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$P(n, n-1)=P(n, n)$$

Use summation notation to write each series. Start the index at \(i=1\). $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{9}$$

Solve each problem. Supports on a Slide\(\quad\) 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.

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$P(n, 1)=n$$

Use summation notation to write each series. Start the index at \(i=1\). $$-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\dots-\frac{1}{2187}$$

Chemical Mixture \(\quad\) 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?

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$P(n, 0)=1$$

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=\frac{n+4}{2 n}$$

Solve each problem. (Modeling) Spending on Food The average family in the United States spends \(\$ 150\) 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?

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, n)=1$$

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=\frac{1+4 n}{2 n}$$

Solve each problem. (Modeling) Growth Pattern for Children 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 \(8 ?\) (IMAGE CANT COPY)

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.

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, 0)=1$$

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=2 e^{n}$$

Concept Check 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.

Sugar Processing A sugar factory receives an order for 1000 units of sugar. The production manager thus orders the production of 1000 units of sugar. He forgets, however, that the production of sugar requires some sugar (to prime the machines, for example), so he ends up with only 900 units of sugar. He then orders an additional 100 units, and receives only 90 units. A further order for 10 units produces 9 units. Finally seeing that he is wrong, the manager decides to try mathematics. He views the production process as an infinite geometric progression with \(a_{1}=1000\) and \(r=0.1 .\) Find the number of units of sugar that he should have ordered originally.

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, 1)=n$$

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

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.

(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.

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, n-1)=n$$

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

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 converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=5-\frac{1}{n}$$

Swing of a Pendulum 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?

Solve each problem. Find the sum of the first six terms of the series $$\frac{\pi^{4}}{90}=\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+\frac{1}{5^{4}}+\cdots+\frac{1}{n^{4}}+\cdots$$ Use your result to estimate \(\pi\). Compare your answer with the actual value of \(\pi\).

Height of a Dropped Ball Beth Schiffer 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.)

Solve each problem. 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 can't copy)

Solve each problem. 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.

(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?

Solve each problem. Refer to Exercise \(85 .\) If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and the growth will slow. According to Verhulst's model, the number of bacteria \(N_{j}\) at time \(40(j-1)\) minutes can be determined by the sequence $$N_{j+1}=\left[\frac{2}{1+\left(N_{j} / K\right)}\right] N_{j}$$, where \(K\) is a constant and \(j \geq 1\). (a) If \(N_{1}=230\) and \(K=5000,\) make a table of \(N_{j}\) for \(j=1,2,3, \ldots, 20 .\) Round values in the table to the nearest integer. (b) Graph the sequence \(N_{j}\) for \(j=1,2,3, \ldots, 20 .\) Use the window \([0,20]\) by \([0,6000]\). (c) Describe the growth of these bacteria when there are limited nutrients. (d) Make a conjecture as to why \(K\) is called the saturation constant. Test your conjecture by changing the value of \(K\) in the given formula.

Solve each problem. The series $$e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\cdots+\frac{a^{n}}{n !}$$ where $$n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \dots \cdot n$$. 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) \(\sqrt{e}\)

Number of Ancestors Suppose a genealogical website allows you to identify all your ancestors that 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.