Chapter 2

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=\frac{1}{4} a_{n}+1\) and \(a_{0}=0\), find \(a_{14}\).

\mathrm{\\{} I n ~ P r o b l e m s ~ 4 3 - 5 0 , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=2, N_{0}=2 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0.1\)

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=2, N_{0}=3 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0.9\)

In Problems \(45-52\), write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{1}{n+1} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=3, N_{0}=1 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{3}{n+3} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=4, N_{0}=2 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.8, x_{0}=0.5\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{n}{n+1} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=\frac{1}{2}, N_{0}=16 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.8, x_{0}=0.1\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{2 n}{(n+2)^{2}} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=\frac{1}{2}, N_{0}=64 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.8, x_{0}=0.9\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{n^{2}+5}{n^{2}+1} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=\frac{1}{3}, N_{0}=81 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.8, x_{0}=0\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{1}{\sqrt{n+1}} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=\frac{1}{4}, N_{0}=16 $$

Adderall is a proprietary combination of amphetamine salts that is used to treat ADHD (AttentionDeficit/Hyperactivity Disorder). The patient takes one pill before \(8 \mathrm{am}\), and the drug has completely entered the blood by \(8 \mathrm{am} .\) At 8 am the blood concentration of the drug is \(33.8 \mathrm{ng} / \mathrm{ml}\). Adderall has first order elimination kinetics with \(7.7 \%\) of the drug being removed from the blood in each hour. The data in this equation are taken from Greenhill et al. (2003). (a) The concentration of drug in the patient's blood \(t\) hours after 8 am is measured to be \(C_{t}\). Write a recursive relation for \(C_{t+1}\) in terms of \(C_{t}\). Assume that the patient takes no other Adderall pills. (b) Solve your recurrence equation to derive an explicit formula for \(C_{t}\) as a function of \(t\). (c) When does the concentration of drug first drop below \(0.1 \mathrm{ng} / \mathrm{ml} ?\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{(-1)^{n+1}}{n+1} $$

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(10 \cdot(0.4)^{t}\). (a) The drug has first order elimination kinetics. \(10 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n^{*}}\) $$ a_{n}=\frac{(-1)^{n}}{n^{3}+3} $$

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(20 \cdot(0.2)^{t}\). (a) The drug has first order elimination kinetics. \(40 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show that, when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.

In Problems \(53-60\), write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=\frac{n^{2}}{n+1} $$

A drug has zeroth order elimination kinetics. At time \(t=0\) an amount \(a_{0}=20 \mathrm{mg}\) is present in the blood. One hour later, at \(t=1\), an amount \(a_{1}=14 \mathrm{mg}\) is present. (a) Assuming that no drug is added to the blood between \(t=0\) and \(t=1\), calculate the amount of drug that is removed from the blood each hour. (b) Write a recursion relation for the amount of drug \(a_{t}\) that is present at time \(t\). Assume no extra drug is added to the blood. (c) Find an explicit formula for \(a_{t}\) as a function of \(t\). (d) When does the amount of drug present in the blood first drop \(\operatorname{to} 0 ?\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=\frac{n+1}{n+2} $$

A drug has first order elimination kinetics. At time \(t=0\) an amount \(a_{0}=20 \mathrm{mg}\) is present in the blood. One hour later, at \(t=1\), an amount \(a_{1}=14 \mathrm{mg}\) is present. (a) Assuming that no drug is added to the blood between \(t=0\) and \(t=1\), calculate the percentage of drug that is removed each hour. (b) Write a recursion relation for the amount of drug \(a_{t}\) present at time \(t\). Assume no extra drug. (c) Find an explicit formula for \(a_{t}\) as a function of \(t\). (d) Will the amount of drug present ever drop to 0 according to vour model?

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=\sqrt{n} $$

You do not know whether a drug has zeroth order or first order elimination kinetics. You will use data to determine which type of kinetics it has. You measure the concentration of the drug (in \(\mathrm{mg} / \mathrm{ml}\) ) at time \(t=0\) and at time \(t=1 .\) No drug is added to the blood between \(t=0\) and \(t=1\). You measure the following data: \begin{tabular}{ll} \hline \(\boldsymbol{t}\) & \(\boldsymbol{c}_{t}\) \\ \hline 0 & 40 \\ 1 & 32 \\ \hline \end{tabular} (a) Assume that the drug has zeroth order kinetics. What amount is eliminated from the blood each hour? (b) Assume that the drug has zeroth order kinetics and no more drug is added to the blood. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (c) Now assume the drug has first order elimination kinetics. What percentage of drug is eliminated from the blood each hour? (d) Assume that the drug has first order kinetics and no more drug is added. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (e) You measure the concentration at time \(t=2\) and find \(c_{2}=\) 25.6. By comparing with your predictions from (b) and (d), decide: Does the drug have zeroth or first order kinetics?

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=n^{2}+3 $$

You do not know whether a drug has zeroth order or first order elimination kinetics. You will use data to determine which type of kinetics it has. You measure the concentration of the drug (in \(\mathrm{mg} / \mathrm{ml}\) ) at time \(t=0\) and at time \(t=1\). No drug is added to the blood in this interval. You measure the following data: \begin{tabular}{ll} \hline \(\boldsymbol{t}\) & \(\boldsymbol{c}_{t}\) \\ \hline 0 & 50 \\ 1 & 35 \\ \hline \end{tabular} (a) Assume that the drug has zeroth order kinetics. What amount is eliminated from the blood each hour? (b) Assume that the drug has zeroth order kinetics and no more drug is added. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (c) Now assume the drug has first order elimination kinetics. What percentage of drug is eliminated from the blood each hour? (d) Assume that the drug has first order kinetics and no more drug is added to the blood. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\) (e) You measure the concentration at time \(t=2\) and find \(c_{2}=\) 20\. By comparing with your predictions from (b) and (d), decide: Does the drug have zeroth or first order kinetics?

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=2^{n} $$

Tylenol in the Body A patient is taking Tylenol (a painkiller that contains acetaminophen) to treat a fever. The data in this question is taken from Rawlins, Henderson, and Hijab (1977). At \(t=0\) the patient takes their first pill. One hour later the drug has been completely absorbed and the blood concentration, measured in \(\mu \mathrm{g} / \mathrm{ml}\), is 15 . Acetaminophen has first order elimination kinetics; in one hour, \(23 \%\) of the acetaminophen present in the blood is eliminated. (a) Write a recursion relation for the concentration \(c_{t}\) of drug in the patient's blood. For \(t \geq 1\) you may assume for now that no other pills are taken after the first one. (b) Find an explicit formula for \(c_{t}\) as a function of \(t\). (c) Suppose that the patient follows the directions on the pill box and takes another Tylenol pill 4 hours after the first (at time \(t=4\) ). What is the concentration at the time at which the second pill is taken? In others words, what is \(c_{4}\) ? (d) Over the next hour \(15 \mathrm{\mug} / \mathrm{ml}\) of drug enter the patient's bloodstream. So, \(c_{5}\) can be calculated from \(c_{4}\) using the word equation: $$ c_{5}=c_{4}+ $$ nt added \(\quad\) amount eliminated blo Given that the amount added is \(15 \mu \mathrm{g} / \mathrm{ml}\), and the amount eliminated is \(0.23 \cdot c_{4}\), calculate \(c_{5} .\) (e) For \(t=5,6,7,8\) the drug continues to be eliminated at a rate of \(23 \%\) per hour. No pills are taken and no extra drug enters the patient's blood. Compute \(c_{8}\). (f) At time \(t=8\), the patient takes another pill. Calculate \(c_{9} .\) Do not forget to include elimination of drug between \(t=8\) and \(t=9\). (g) We want to calculate the maximum concentration of drug in the patient's blood. We know that concentrations are highest in the hour after a pill is taken, namely at time \(t=1, t=5, t=\) \(9, \ldots\) Define a sequence \(C_{n}\) representing the concentration of the drug one hour after the \(n\) th pill is taken. (h) What terms of the original sequence \(\left\\{c_{r}: t=1,2, \ldots\right\\}\) are \(C_{1}\), \(C_{2}\), and \(C_{3} ?\) (i) Explain why $$ C_{n+1}=(0.77)^{4} \cdot C_{n}+15 $$ and \(c_{1}=15\) (j) From the recursion relation, assuming that the patient continues to take Tylenol pills at 4 -hour intervals, calculate \(C_{1}, C_{2}\), \(C_{3}, C_{4}, C_{5}\), and \(C_{6}\) (k) Does \(C_{n}\) increase indefinitely, or do you think that it converges? (1) By looking for fixing point of the recursion relation in (h), find the limit of \(C_{n}\) as \(n \rightarrow \infty\).

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=2^{n+3} $$

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements for drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{c}_{t} \text { (mg/liter) } & 16 & 12 & 9 & 6.75 \\ \hline \end{array} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=3^{n} $$

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements of drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 1 & 2 & 3 & 4 \\ \hline c_{t}(\mu \mathrm{g} / \mathrm{m} \mathrm{l}) & 20 & 18 & 16 & 14 \\ \hline \end{array} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=3^{-2 n} $$

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements of drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 1 & 2 & 3 & 4 \\ \hline c_{t}(\mu \mathrm{g} / \mathrm{ml}) & 40 & 36 & 32 & 28 \\ \hline \end{array} $$

In Problems 61-72, \(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{n}, \epsilon=0.01 $$

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements for drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 0 & 2 & 4 & 6 \\ \hline \boldsymbol{c}_{t}(\mathbf{m g} / \mathbf{m l}) & 40 & 36 & 32.4 & 29.16 \\ \hline \end{array} $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{n}, \epsilon=0.02 $$

62\. You are an ecologist measuring the size of the population of plants living on an island. You measure the size of the population each year at the beginning of the breeding season. Denote the population size after \(t\) years by \(N_{t}\). The first year that you measure the population size there are 16 birds, that is \(N_{1}=16\). Each year you find \(50 \%\) more birds than the year before. (a) Write down the recursion for \(N_{t}\) and solve it. (b) According to your formula, when will the size of the population first exceed 100 birds? (c) When will the population size first exceed 1000 birds? (d) Use your formula for \(N_{t}\) to predict when the population size will first exceed one million birds. Should you believe this prediction?

Hormone Implant You are studying an implanted contraceptive that releases hormone continuously into a patient's blood. The data in this question are from Nilsson et al. (1986). The device adds \(20 \mu \mathrm{g}\) of hormone to the blood each day. In the blood the hormone has first order elimination kinetics; \(4 \%\) of the hormone is eliminated each day. (a) Let the amount of hormone in the blood on day \(t\) be \(a_{t} .\) Write a word equation for the change in \(a_{t}\) over one day. (b) Put in mathematical formulas for each of the terms in your word equation from (a). (c) Assume that on day 0 no hormone is present in the patient's blood, in other words, \(a_{0}=0 .\) Use your equation from (b) to compute the amount of hormone in the blood on days \(1,2,3,4\), \(5,6 .\) (d) Over time the level of hormone in the blood converges to a limit. Find the value of this limit by looking for a fixed point of your recurrence relation in (b).

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.01 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.001 $$