Excursions in Modern Mathematics

Peter Tannenbaum

Chapter 9 Population Growth Models - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider the sequence defined by the explicit formula $A_{N}=N^{2}+1$
(a) Find $A_{1}$.
(b) Find $A_{100}$.
(c) Suppose $A_{N}=10 .$ Find $N$.

Ankit Gupta

Numerade Educator

01:12

Problem 2

Consider the sequence defined by the explicit formula $A_{N}=3^{N}-2$
(a) Find $A_{3}$.
(b) Use a calculator to find $A_{15}$.
(c) Suppose $A_{N}=79 .$ Find $N$.

William Scherer

Numerade Educator

01:54

Problem 3

Consider the sequence defined by the explicit formula $A_{N}=\frac{4 N}{N+3}$
(a) Find $A_{1}$.
(b) Find $A_{9}$.
(c) Suppose $A_{N}=\frac{5}{2},$ Find $N$.

Julie Silva

Numerade Educator

01:57

Problem 4

Consider the sequence defined by the explicit formula $A_{N}=\frac{2 N+3}{3 N-1}$
(a) Find $A_{1}$.
(b) Find $A_{100}$ -
(c) Suppose $A_{N}=1 .$ Find $N$.

Aman Gupta

Numerade Educator

01:57

Problem 5

Consider the sequence defined by the explicit formula $A_{N}=(-1)^{N+1}$
(a) Find $A_{1}$ -
(b) Find $A_{100}$ -
(c) Find all values of $N$ for which $A_{N}=1$.

Aman Gupta

Numerade Educator

01:49

Problem 6

Consider the sequence defined by the explicit formula $A_{N}=\left(-\frac{1}{N}\right)^{N-1}$
(a) Find $A_{1}$.
(b) Find $A_{4}$
(c) Find all values of $N$ for which $A_{N}$ is positive.

Julie Silva

Numerade Educator

00:55

Problem 7

Consider the sequence defined by the recursive formula $A_{N}=2 A_{N-1}+A_{N-2}$ and starting with $A_{1}=1, A_{2}=1$
(a) List the next four terms of the sequence.
(b) Find $A_{8}$.

Amy Jiang

Numerade Educator

00:55

Problem 8

Consider the sequence defined by the recursive formula $A_{N}=A_{N-1}+2 A_{N-2}$ and starting with $A_{1}=1, A_{2}=1$
(a) List the next four terms of the sequence.
(b) Find $A_{8}$.

Amy Jiang

Numerade Educator

00:55

Problem 9

Consider the sequence defined by the recursive formula $A_{N}=A_{N-1}-2 A_{N-2}$ and starting with $A_{1}=1, A_{2}=-1$
(a) List the next four terms of the sequence.
(b) Find $A_{8}$.

Amy Jiang

Numerade Educator

02:26

Problem 10

Consider the sequence defined by the recursive formula $A_{N}=2 A_{N-1}-3 A_{N-2}$ and starting with $A_{1}=-1, A_{2}=1$
(a) List the next four terms of the sequence.
(b) Find $A_{8}$.

Lourence Gonhovi

Numerade Educator

00:27

Consider the sequence $1,4,9,16,25, \ldots$
(a) List the next two terms of the sequence.
(b) Assuming the sequence is denoted by $A_{1}, A_{2}, A_{3}, \ldots$ give an explicit formula for $A_{N}$.
(c) Assuming the sequence is denoted by $P_{0}, P_{1}, P_{2}, \ldots$ give an explicit formula for $P_{N}$

Trinity Steen

Numerade Educator

02:06

Problem 12

Consider the sequence $1,2,6,24,120, \ldots$
(a) List the next two terms of the sequence.
(b) Assuming the sequence is denoted by $A_{1}, A_{2}, A_{3}, \ldots,$ give an explicit formula for $A_{N}$.
(c) Assuming the sequence is denoted by $P_{0}, P_{1}, P_{2}, \ldots$, give an explicit formula for $P_{N}$.

Linh Vu

Numerade Educator

02:06

Problem 13

Consider the sequence $0,1,3,6,10,15,21, \ldots$
(a) List the next two terms of the sequence.
(b) Assuming the sequence is denoted by $A_{1}, A_{2}, A_{3}, \ldots$ give an explicit formula for $A_{N}$.
(c) Assuming the sequence is denoted by $P_{0}, P_{1}, P_{2}, \ldots$, give an explicit formula for $P_{N}$.

Linh Vu

Numerade Educator

02:06

Problem 14

Consider the sequence $2,3,5,9,17,33, \ldots$
(a) List the next two terms of the sequence.
(b) Assuming the sequence is denoted by $A_{1}, A_{2}, A_{3}, \ldots$, give an explicit formula for $A_{N}$.
(c) Assuming the sequence is denoted by $P_{0}, P_{1}, P_{2}, \ldots$. give an explicit formula for $P_{N}$.

Linh Vu

Numerade Educator

01:54

Problem 15

Consider the sequence $1, \frac{8}{5}, 2, \frac{16}{7}, \frac{20}{8}, \ldots$
(a) List the next two terms of the sequence.
(b) If the notation for the sequence is $A_{1}, A_{2}, A_{3}, \ldots,$ give an explicit formula for $A_{N}$.

Linh Vu

Numerade Educator

01:02

Problem 16

Consider the sequence $3,2, \frac{5}{4}, \frac{6}{8}, \frac{7}{16}, \ldots$
(a) List the next two terms of the sequence.
(b) If the notation for the sequence is $A_{1}, A_{2}, A_{3}, \ldots,$ give an explicit formula for $A_{N}$

Nick Johnson

Numerade Educator

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Problem 17

Airlines would like to board passengers in the order of decreasing seat numbers (largest seat number first, second largest next, and so on), but passengers don't like this policy and refuse to go along. If two passengers randomly board a plane the probability that they board in order of decreasing seat numbers is $\frac{1}{2}$, if three passengers randomly board a plane the probability that they board in order of decreasing seat numbers is $\frac{1}{6} ;$ if four passengers randomly board a plane the probability that they board in order of decreasing seat numbers is $\frac{1}{24}$; and if five passengers randomly board a plane, the probability that they board in order of decreasing seat numbers is $\frac{1}{120}$. Using the sequence $\frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots$ as your guide,
(a) determine the probability that if six passengers randomly board a plane they board in order of decreasing seat numbers.
(b) determine the probability that if 12 passengers randomly board a plane they board in order of decreasing seat numbers

Joshua Argo

Numerade Educator

02:16

Problem 18

18. When two fair coins are tossed the probability of tossing two heads is $\frac{1}{4}$, when three fair coins are tossed the probability of tossing two heads and one tail is $\frac{3}{8} ;$ when four fair coins are tossed the probability of tossing two heads and two tails is $\frac{6}{16}$ when five fair coins are tossed the probability of tossing two heads and three tails is $\frac{10}{32}$. Using the sequence $\frac{1}{4}, \frac{3}{8}, \frac{6}{16}, \frac{10}{32}, \ldots$ as your guide.
(a) determine the probability of tossing two heads and four tails when six fair coins are tossed.
(b) determine the probability of tossing two heads and 10 tails when 12 fair coins are tossed. (Hint: Find an explicit formula first.)

Joe Lesueur

Numerade Educator

07:17

Problem 19

Consider a population that grows linearly following the recursive formula $P_{N}=P_{N-1}+125$, with initial population $P_{0}=80 .$
(a) Find $P_{1}, P_{2}$, and $P_{3}$.
(b) Give an explicit formula for $P_{N}$.
(c) Find $P_{100}$.

Aishwarya Krishnakumar

Numerade Educator

05:14

Problem 20

Consider a population that grows linearly following the recursive formula $P_{N}=P_{N-1}+23,$ with initial population $P_{0}=57$
(a) Find $P_{1}, P_{2},$ and $P_{3}$
(b) Give an explicit formula for $P_{N}$.
(c) Find $P_{200}$

Ryan Williams

Numerade Educator

05:14

Problem 21

Consider a population that grows linearly following the recursive formula $P_{N}=P_{N-1}-25,$ with initial population $P_{0}=578$
(a) Find $P_{1}, P_{2},$ and $P_{3}$.
(b) Give an explicit formula for $P_{N}$.
(c) Find $P_{23}$.

Ryan Williams

Numerade Educator

02:21

Problem 22

Consider a population that grows linearly following the recursive formula $P_{N}=P_{N-1}-111,$ with initial population $P_{0}=11,111$
(a) Find $P_{1}, P_{2},$ and $P_{3}$,
(b) Give an explicit formula for $P_{N}$.
(c) Find $P_{100}$.

Lourence Gonhovi

Numerade Educator

03:42

Problem 23

Consider a population that grows linearly, with $P_{0}=8$ and $P_{10}=38$
(a) Give an explicit formula for $P_{N}$.
(b) Find $P_{50}$.

Clarissa Noh

Numerade Educator

00:21

Problem 24

Consider a population that grows linearly, with $P_{5}=37$ and $P_{7}=47$
(a) Find $P_{0}$.
(b) Give an explicit formula for $P_{N}$.
(c) Find $P_{100}$.

James Kiss

Numerade Educator

01:19

Problem 25

Official unemployment rates for the U.S. population are reported on a monthly basis by the Bureau of Labor Statistics. For the period October, $2011,$ through January, $2012,$ the official unemployment rates were $8.9 \%$ (Oct.), $8.7 \%$ (Nov.), $8.5 \%$ (Dec.), and 8.3\% (Jan.). (Source: U.S. Bureau of Labor Statistics, www.bls.gov.) If the unemployment rates were to continue to decrease following a linear model,
(a) predict the unemployment rate on January, $2013 .$
(b) predict when the United States would reach a zero unemplovment rate

Doris Bennett

Numerade Educator

05:47

Problem 26

The world population reached 6 billion people in 1999 and 7 billion in 2012. (Source: Negative Population Growth, www.npg. org.) Assuming a linear growth model for the world population.
(a) predict the year when the world population would reach 8 billion.
(b) predict the world population in 2020 .

Anthony Ramos

Numerade Educator

04:17

Problem 27

The Social Security Administration uses a linear growth model to estimate life expectancy in the United States. The model uses the explicit formula $L_{N}=66.17+0.96 N$ where $L_{N}$ is the life expectancy of a person born in the year $1995+N$ (i.e., $N=0$ corresponds to 1995 as the year of birth, $N=1$ corresponds to 1996 as the year of birth, and so on). (Source: Social Security Administration, www.socialsecurity.gov.)
(a) Assuming the model continues to work indefinitely, estimate the life expectancy of a person born in $2012 .$
(b) Assuming the model continues to work indefinitely, what year will you have to be born so that your life expectancy is $90 ?$

Eric Mockensturm

Numerade Educator

03:09

Problem 28

While the number of smokers for the general adult population is decreasing, it is not decreasing equally across all subpopulations (and for some groups it is actually increasing). For the 18 -to- 24 age group, the number of smokers was 8 million in 1965 and 6.3 million in $2009 .$ (Source: "Trends in Tobacco Use," American Lung Association, 2011.) Assuming the number of smokers in the $18-$ to- 24 age group continues decreasing according to a negative linear growth model,
(a) predict the number of smokers in the $18-$ to -24 age group in 2015 (round your answer to the nearest thousand).
(b) predict in what year the number of smokers in the 18 to- 24 age group will reach 5 million.

Joseph Lentino

Numerade Educator

02:15

Problem 29

Use the arithmetic sum formula to find the sum $\underbrace{2+7+12+\cdots+497}_{100 \text { terms }}$

Ankit Gupta

Numerade Educator

01:51

Problem 30

Use the arithmetic sum formula to find the sunn $\underbrace{21+28+35+\cdots+413}_{57 \text { terms }}$

Nicholas Bondra

Numerade Educator

00:33

Problem 31

An arithmetic sequence has first term $P_{0}=12$ and common difference $d=3$
(a) The number 309 is which term of the arithmetic sequence?
(b) Find the $\operatorname{sum} 12+15+18+\cdots+309$.

Angela Guo

Numerade Educator

01:02

Problem 32

An arithmetic sequence has first term $P_{0}=1$ and common difference $d=9$
(a) The number 2701 is which term of the arithmetic sequence?
(b) Find $1+10+19+\cdots+2701$.

Amy Jiang

Numerade Educator

00:57

Problem 33

Find the sum
(a) $1+3+5+7+\cdots+149 .$ (Hint: See Example 9.13.
(b) $\underbrace{1+3+5+\cdots}_{100 \text { terms }}$

Fuzail Shakir

Numerade Educator

01:56

Problem 34

Find the sum
(a) $2+4+6+\cdots+98$
(b) $\underbrace{2+4+6+\cdots}_{75 \text { terms }}$

Julie Silva

Numerade Educator

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Problem 35

The city of Lightsville currently has 137 streetlights. As part of an urban renewal program, the city council has decided to install and have operational 2 additional streetlights at the end of each week for the next 52 weeks. Each streetlight costs $\$ 1$ to operate for 1 week.
(a) How many streetlights will the city have at the end of 38 weeks?
(b) How many streetlights will the city have at the end of $N$ weeks? (Assume $N \leq 52 .)$
(c) What is the cost of operating the original 137 lights for 52 weeks?
(d) What is the additional cost for operating the newly installed lights for the 52 -week period during which they are being installed?

Alison Rodriguez

Numerade Educator

04:02

Problem 36

A manufacturer currently has on hand 387 widgets. During the next 2 years, the manufacturer will be increasing his inventory by 37 widgets per week. (Assume that there are exactly 52 weeks in one year.) Each widget costs 10 cents a week to store.
(a) How many widgets will the manufacturer have on hand after 20 weeks?
(b) How many widgets will the manufacturer have on hand after $N$ weeks? (Assume $N \leq 104$.)
(c) What is the cost of storing the original 387 widgets for 2 years (104 weeks)?
(d) What is the additional cost of storing the increased inventory of widgets for the next 2 years?

Zach Steedman

Numerade Educator

07:17

Problem 37

A population grows according to an exponential growth model. The initial population is $P_{0}=11$ and the common ratio is $R=1.25 .$
(a) Find $P_{1}$.
(b) Find $P_{9}$.
(c) Give an explicit formula for $P_{N}$.

Aishwarya Krishnakumar

Numerade Educator

02:09

Problem 38

A population grows according to an exponential growth model, with $P_{0}=8$ and $P_{1}=12$
(a) Find the common ratio $R$.
(b) Find $P_{9}$.
(c) Give an explicit formula for $P_{N}$.

Vishal Parmar

Numerade Educator

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Problem 39

A population grows according to the recursive rule $P_{N}=4 P_{N-1},$ with initial population $P_{0}=5 .$
(a) Find $P_{1}, P_{2},$ and $P_{3}$.
(b) Give an explicit formula for $P_{N}$.
(c) How many generations will it take for the population to reach 1 million?

Clayton Bennett

Numerade Educator

07:17

Problem 40

A population decays according to an exponential growth model, with $P_{0}=3072$ and common ratio $R=0.75 .$
(a) Find $P_{5}$.
(b) Give an explicit formula for $P_{N}$.
(c) How many generations will it take for the population

Aishwarya Krishnakumar

Numerade Educator

01:15

Problem 41

Crime in Happyville is on the rise. Each year the number of crimes committed increases by $50 \%$. Assume that there were 200 crimes committed in $2010,$ and let $P_{\mathrm{N}}$ denote the number of crimes committed in the year $2010+N$.
(a) Give a recursive description of $P_{N}$
(b) Give an explicit description of $P_{N}$
(c) If the trend continues, approximately how many crimes will be committed in Happyville in the year $2020 ?$

Carson Merrill

Numerade Educator

04:46

Problem 42

Since $2010,$ when 100,000 cases were reported, each year the number of new cases of equine flu has decreased by $20 \% .$ Let $P_{N}$ denote the number of new cases of equine flu in the year $2010+N$
(a) Give a recursive description of $P_{N}$
(b) Give an explicit description of $P_{N}$.
(c) If the trend continues, approximately how many new cases of equine flu will be reported in the year $2025 ?$

Abhishek Kumar

Numerade Educator

01:49

Problem 43

In 2010 the number of mathematics majors at Bright State University was 425 ; in 2011 the number of mathematics majors was 463 . Find the growth rate (expressed as a percent) of mathematics majors from 2010 to 2011 .

James Kiss

Numerade Educator

06:04

Problem 44

Avian influenza A(H5N1) is a particularly virulent strain of the bird flu. In 2008 there were 44 cases of avian influenza A(H5N1) confirmed worldwide; in 2009 the number of confirmed cases worldwide was 73. (Source: World Health Organization, www.who.int.) Find the growth rate in the number of confirmed cases worldwide of avian influenza A(H5N1) from 2008 to 2009 . Express your answer as a percent.

Bobby Barnes

University of North Texas

02:47

Problem 45

In 2010 the undergraduate enrollment at Bright State University was 19,753 ; in 2011 the undergraduate enrollment was $17,389 .$ Find the "growth" rate in the undergraduate enrollment from 2010 to 2011. Give your answer as a percent.

Craydon Maloney

Numerade Educator

06:04

Problem 46

cases worldwide was 48. (Source: World Health Organization, www, who.int.) Find the "growth" rate in the number of confirmed cases worldwide of avian influenza A(H5N1). Give your answer as a percent.

Bobby Barnes

University of North Texas

00:29

Problem 48

Consider the geometric sequence $P_{0}=4, P_{1}=6, P_{2}=9, \ldots$
(a) Find the common ratio $R$.
(b) Use the geometric sum formula to find the sum $P_{0}+P_{1}+\cdots+P_{24}$

James Kiss

Numerade Educator

01:17

Problem 50

Consider the geometric sequence $P_{0}=10, P_{1}=2, P_{2}=$ $0.4 \ldots$
(a) Find the common ratio $R$.
(b) Use the geometric sum formula to find the sum $P_{0}+P_{1}+\cdots+P_{24}$

Sheryl Ezze

Numerade Educator

02:32

Problem 51

Find the sum
(a) $1+2+2^{2}+2^{3}+\cdots+2^{15}$
(b) $1+2+2^{2}+2^{3}+\cdots+2^{N-1}$. (Hint: The answer is an expression in $N$.)

Mir Afzal

Numerade Educator

02:32

Problem 52

Find the sum
(a) $1+3+3^{2}+3^{3}+\cdots+3^{10}$
(b) $1+3+3^{2}+3^{3}+\cdots+3^{N-1}$. (Hint: The answer is an expression in $N$.)

Mir Afzal

Numerade Educator

02:29

Problem 53

A population grows according to the logistic growth model, with growth parameter $r=0.8$. Starting with an initial population given by $p_{0}=0.3,$
(a) find $p_{1}$.
(b) find $p_{2}$.
(c) determine what percent of the habitat's carrying capacity is taken up by the third generation.

Monica Miller

Numerade Educator