## MATH1013 Quizzes

Quiz 4: First order linear difference equations
Question 1 Questions
What are the next four terms in the sequence ${x}_{n+1}-0.4{x}_{n}-150=0$, where ${x}_{6}^{}=50\phantom{\rule{0.3em}{0ex}}?$
 a) ${x}_{1}^{}=170\phantom{\rule{0.3em}{0ex}}$, ${x}_{2}^{}=218\phantom{\rule{0.3em}{0ex}}$, ${x}_{3}^{}=237.2\phantom{\rule{0.3em}{0ex}}$, ${x}_{4}^{}=244.88$ b) ${x}_{1}^{}=210\phantom{\rule{0.3em}{0ex}}$, ${x}_{2}^{}=234\phantom{\rule{0.3em}{0ex}}$, ${x}_{3}^{}=243.6\phantom{\rule{0.3em}{0ex}}$, ${x}_{4}^{}=247.44$ c) ${x}_{7}^{}=170\phantom{\rule{0.3em}{0ex}}$, ${x}_{8}^{}=218\phantom{\rule{0.3em}{0ex}}$, ${x}_{9}^{}=237.2\phantom{\rule{0.3em}{0ex}}$, ${x}_{10}^{}=244.88$ d) ${x}_{7}^{}=210\phantom{\rule{0.3em}{0ex}}$, ${x}_{8}^{}=234\phantom{\rule{0.3em}{0ex}}$, ${x}_{9}^{}=243.6\phantom{\rule{0.3em}{0ex}}$, ${x}_{10}^{}=247.44$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
${x}_{n+1}=0.4{x}_{n}+150\phantom{\rule{0.3em}{0ex}}.$ Since we are given ${x}_{6}^{}\phantom{\rule{0.3em}{0ex}},$ the next 4 terms are $\begin{array}{llll}\hfill {x}_{7}^{}& =0.4\ast 50+150=170,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {x}_{8}^{}& =0.4\ast 170+150=218,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {x}_{9}^{}& =0.4\ast 218+150=237.2,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {x}_{7}^{}& =0.4\ast 237.2+150=244.80\phantom{\rule{1em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (d) is incorrect
What are the next four terms in the sequence ${u}_{n+2}-5{u}_{n+1}+6{u}_{n}=20$, where ${u}_{1}^{}=4$ and ${u}_{2}^{}=9\phantom{\rule{0.3em}{0ex}}?$
 a) ${u}_{2}^{}=9\phantom{\rule{0.3em}{0ex}}$, ${u}_{3}^{}=41\phantom{\rule{0.3em}{0ex}}$, ${x}_{4}^{}=171\phantom{\rule{0.3em}{0ex}}$, ${u}_{5}^{}=629$ b) ${u}_{3}^{}=41\phantom{\rule{0.3em}{0ex}}$, ${x}_{4}^{}=171\phantom{\rule{0.3em}{0ex}}$, ${u}_{5}^{}=629\phantom{\rule{0.3em}{0ex}},{u}_{6}^{}=2139$ c) ${u}_{2}^{}=9\phantom{\rule{0.3em}{0ex}}$, ${u}_{3}^{}=14\phantom{\rule{0.3em}{0ex}}$, ${x}_{4}^{}=74\phantom{\rule{0.3em}{0ex}}$, ${u}_{5}^{}=308$ d) ${u}_{3}^{}=14\phantom{\rule{0.3em}{0ex}}$, ${x}_{4}^{}=74\phantom{\rule{0.3em}{0ex}}$, ${u}_{5}^{}=308\phantom{\rule{0.3em}{0ex}}$, ${u}_{6}^{}=1116$

Choice (a) is incorrect
Choice (b) is correct!
${u}_{n+2}=5{u}_{n+1}-6{u}_{n}+20\phantom{\rule{0.3em}{0ex}}.$ Since we are given ${u}_{1}^{}$ and ${u}_{2}^{}$ the next 4 terms are $\begin{array}{llll}\hfill {u}_{3}^{}& =5\ast 9-6\ast 4+20=41,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {u}_{4}^{}& =5\ast 41-6\ast 9+20=171,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {u}_{5}^{}& =5\ast 171-6\ast 41+20=629,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {u}_{6}^{}& =5\ast 629-6\ast 171+20=2139\phantom{\rule{1em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Let ${C}_{n}$ be the number of people who were on the subscription list of the magazine “UFO today” $n$ years after the beginning of 1995. What is the meaning of ${C}_{4}\phantom{\rule{0.3em}{0ex}}?$
 a) ${C}_{4}=$ the number of people who were on the subscription list of the magazine in the year 2000. b) ${C}_{4}=$ the number of people who were on the subscription list of the magazine at the beginning of the year 2000. c) ${C}_{4}=$ the number of people who were on the subscription list of the magazine in 1998. d) ${C}_{4}=$ the number of people who were on the subscription list of the magazine at the beginning of 1999.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
${C}_{1}$ is the number of people who were on the subscription list of the magazine at the beginning of 1996.
${C}_{2}$ is the number of people who were on the subscription list of the magazine at the beginning of 1997.
${C}_{3}$ is the number of people who were on the subscription list of the magazine at the beginning of 1998.
${C}_{4}$ is the number of people who were on the subscription list of the magazine at the beginning of 1999.
Suppose ${P}_{n}=0.4{P}_{n-1}$ and ${P}_{1}=100\phantom{\rule{0.3em}{0ex}}.$ What is ${P}_{n}$ in terms of $n\phantom{\rule{0.3em}{0ex}}?$
 a) ${P}_{n}=100{\left(0.4\right)}^{n-1}$ b) ${P}_{n}=100{\left(0.4\right)}^{n}$ c) ${P}_{n}=100{\left(0.4\right)}^{n-1}\left(0.6\right)$ d) ${P}_{n}=100{\left(0.6\right)}^{n-1}\left(0.4\right)$

Choice (a) is correct!
If ${P}_{n}=a{P}_{n-1}$ then ${P}_{n}={a}^{n-1}{P}_{1}\phantom{\rule{0.3em}{0ex}}.$
Therefore ${P}_{n}={\left(0.4\right)}^{n-1}100\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Suppose ${P}_{n}=0.8{P}_{n-1}\phantom{\rule{0.3em}{0ex}},$ where $n$ is the number of years since 1960. Determine how long it would take for the population to halve.
 a) It would take until 1964. b) Just over 3 years. c) Just over 4 years. d) The answer is negative and makes no sense.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
We wish to solve ${P}_{n}=\frac{1}{2}{P}_{1}={\left(0.8\right)}^{n-1}{P}_{1}\phantom{\rule{0.3em}{0ex}},$ therefore $\begin{array}{llll}\hfill {\left(0.8\right)}^{n-1}& =\frac{1}{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln{\left(0.8\right)}^{n-1}& =ln\frac{1}{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \left(n-1\right)ln\left(0.8\right)& =-ln2,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill n-1& =-\frac{ln2}{ln0.8}=3.106,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill n& =4.106\phantom{\rule{1em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (d) is incorrect
Suppose ${P}_{n}=1.04{P}_{n-1}\phantom{\rule{0.3em}{0ex}},$ where $n$ is the number of years since 1980. Determine how long it would take for the population to double.
 a) It would take until 2008. b) Just over 2 years. c) About 17 years and 8 months. d) About 18 years and 8 months.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
We wish to solve ${P}_{n}=2{P}_{1}={\left(1.04\right)}^{n-1}{P}_{1}\phantom{\rule{0.3em}{0ex}},$ therefore $\begin{array}{llll}\hfill {\left(1.04\right)}^{n-1}& =2,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln{\left(1.04\right)}^{n-1}& =ln2,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \left(n-1\right)ln\left(1.04\right)& =ln2,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill n-1& =\frac{ln2}{ln1.04}=17.673,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill n& =18.673\phantom{\rule{1em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Questions 7 and 8 use the same information.

The Bureau of Statistics has issued the following annual population information for the town of Agga Agga for the ten years since 1990. The town has an annual birth rate of $4.5%$ of the population and an annual death rate of $2.5%$ and there was a nett inflow of $100$ people migrating to the town each year.

Which of the difference equations below would model the population, ${P}_{n}$, of Agga Agga where $n$ is the number of years since 1990 ?
 a) ${P}_{n}=7{P}_{n-1}-100$ b) ${P}_{n}-2{P}_{n-1}-100=0$ c) ${P}_{n}-1.02{P}_{n-1}-100=0$ d) ${P}_{n}=1.07{P}_{n-1}-100$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The nett growth rate is $2%\phantom{\rule{0.3em}{0ex}}.$ If there was no migration we would have ${P}_{n}=1.02{P}_{n-1}\phantom{\rule{0.3em}{0ex}}.$ With the nett migration of 100 people we would have ${P}_{n}=1.02{P}_{n-1}+100\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Questions 7 and 8 use the same information.

The Bureau of Statistics has issued the following annual population information for the town of Agga Agga for the ten years since 1990. The town has an annual birth rate of $4.5%$ of the population and an annual death rate of $2.5%$ and there was a nett inflow of $100$ people migrating to the town each year.

If the population of Agga Agga was 10 000 in 1990 what was the population to the nearest 10 people in 1995 ?
 a) $11\phantom{\rule{0.3em}{0ex}}240$ b) $11\phantom{\rule{0.3em}{0ex}}560$ c) $11\phantom{\rule{0.3em}{0ex}}890$ d) $10\phantom{\rule{0.3em}{0ex}}520$

Choice (a) is incorrect
Choice (b) is correct!
If ${P}_{n}$ is the population of Agga Agga then ${P}_{0}=10\phantom{\rule{0.3em}{0ex}}000$ so we are looking for ${P}_{5}$ and
${P}_{1}=10\phantom{\rule{0.3em}{0ex}}300\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}{P}_{2}=10\phantom{\rule{0.3em}{0ex}}606\phantom{\rule{0.3em}{0ex}},{P}_{3}=10\phantom{\rule{0.3em}{0ex}}918.12\phantom{\rule{0.3em}{0ex}},{P}_{4}=11\phantom{\rule{0.3em}{0ex}}236.21\phantom{\rule{0.3em}{0ex}},{P}_{5}=11\phantom{\rule{0.3em}{0ex}}561.21\phantom{\rule{1em}{0ex}}.$
Choice (c) is incorrect
Choice (d) is incorrect
Questions 9 and 10 use the same information.

A population of Bilbies in western New South Wales is decreasing at a rate of $8%$ per year. The population has been supplemented by 75 breeding pairs per year from Taronga Zoo starting in 1998.

Which of the following difference equations model the population if ${P}_{n}$ is the population $n$ years since the population supplement began ?
 a) ${P}_{n}-0.92{P}_{n-1}-150=0$ b) ${P}_{n}-1.08{P}_{n-1}-75=0$ c) ${P}_{n}-1.08{P}_{n-1}-150=0$ d) ${P}_{n}-0.92{P}_{n-1}-75=0$

Choice (a) is correct!
The rate of population decrease is $8%\phantom{\rule{0.3em}{0ex}},$ i.e there is only $92%$ of the population left after each year. If there was no supplement we would have ${P}_{n}=0.92{P}_{n-1}\phantom{\rule{0.3em}{0ex}}.$ With the supplement of 150 Bilbies we would have ${P}_{n}=0.92{P}_{n-1}+150\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Questions 9 and 10 use the same information.

A population of Bilbies in western New South Wales is decreasing at a rate of $8%$ per year. The population has been supplemented by 75 breeding pairs per year from Taronga Zoo starting in 1998.

If there were 200 Bilbies in 1999 what would the population of the Bilbies be in 2004 ?
 a) 859 b) 451 c) 490 d) 771

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
If ${P}_{n}$ is the population of the Bilbies $n$ years after the breeding program began then ${P}_{1}=200$ in 1999 so we are looking for ${P}_{6}$ in 2004 and
${P}_{2}=334\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}{P}_{3}=457.28\phantom{\rule{0.3em}{0ex}},{P}_{4}=570.7\phantom{\rule{0.3em}{0ex}},{P}_{5}=675.04\phantom{\rule{0.3em}{0ex}},{P}_{6}=771.04\phantom{\rule{0.3em}{0ex}}.$