What are the next four terms in the sequence
${x}_{n+1}-0.4{x}_{n}-150=0$,
where ${x}_{6}^{}=50\phantom{\rule{0.3em}{0ex}}?$

*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 ${u}_{2}^{}=9\phantom{\rule{0.3em}{0ex}}?$

*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
${C}_{4}\phantom{\rule{0.3em}{0ex}}?$

*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.

${C}_{2}$ 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 $n\phantom{\rule{0.3em}{0ex}}?$

*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 ${P}_{n}={\left(0.4\right)}^{n-1}100\phantom{\rule{0.3em}{0ex}}.$

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.

*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.

*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 ?

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 ?

*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 ?

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 ?

*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 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 ?

*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 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 ?

*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

$${P}_{2}=334\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}{P}_{3}=457.28\phantom{\rule{0.3em}{0ex}},{P}_{4}=570.7\phantom{\rule{0.3em}{0ex}},{P}_{5}=675.04\phantom{\rule{0.3em}{0ex}},{P}_{6}=771.04\phantom{\rule{0.3em}{0ex}}.$$