John purchases a new bicycle for 880 US dollars (USD) and pays for it with a Canadian credit card. There is a transaction fee of 4.2 % charged to John by the credit card company to convert this purchase into Canadian dollars (CAD).

The exchange rate is 1 USD = 1.25 CAD.

John insures his bicycle with a US company. The insurance company produces the following table for the bicycle’s value during each year.

The values of the bicycle form a geometric sequence.

During the 1st year John pays 120 USD to insure his bicycle. Each year the amount he pays to insure his bicycle is reduced by 3.50 USD.

A large underground tank is constructed at Mills Airport to store fuel. The tank is in the shape of an isosceles trapezoidal prism, $\text{ABCDEFGH}$.

$\text{AB}=70 \text{m}$ , $\text{AF}=200 \text{m}$, $\text{AD}=40 \text{m}$, $\text{BC}=40 \text{m}$ and $\text{CD}=110 \text{m}$. Angle $\text{ADC}=60°$ and angle $\text{BCD}=60°$. The tank is illustrated below.

Once construction was complete, a fuel pump was used to pump fuel into the empty tank. The amount of fuel pumped into the tank by this pump each hour decreases as an arithmetic sequence with terms $u_1, u_2, u_3, \dots , u_n$.

Part of this sequence is shown in the table.

At the end of the $2\text{nd}$ hour, the total volume of fuel in the tank was $88 200 {\text{m}}^3$.

Two friends Amelia and Bill, each set themselves a target of saving $\$20 000$. They each have $\$9000$ to invest.

Amelia invests her $\$9000$ in an account that offers an interest rate of $7\%$ per annum compounded annually.

A third friend Chris also wants to reach the $\$20 000$ target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.

Tommaso plans to compete in a regional bicycle race after he graduates, however he needs to buy a racing bicycle. He finds a bicycle that costs 1100 euro (EUR). Tommaso has 950 EUR and invests this money in an account that pays 5 % interest per year, compounded monthly.

The cost of the bicycle, $C$, can be modelled by $C=20x+1100$, where $x$ is the number of years since Tommaso invested his money.

An arithmetic sequence has first term $60$ and common difference $-2.5$.

A new café opened and during the first week their profit was $60.

The café’s profit increases by $10 every week.

A new tea-shop opened at the same time as the café. During the first week their profit was also $60.

The tea-shop’s profit increases by 10 % every week.

In this question, give all answers correct to two decimal places.

Sam invests $\$1700$ in a savings account that pays a nominal annual rate of interest of $2.74\%$, compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account.

David also invests $\$1700$ in a savings account that pays an annual rate of interest of $r\%$, compounded yearly. David makes no further payments or withdrawals from this account.

The following table shows values of ln x and ln y.

The relationship between ln x and ln y can be modelled by the regression equation ln y = a ln x + b.

The sum of the first $n$ terms of a geometric sequence is given by $S_n=\underset{r=1}{\overset{n}{\text{Σ}}}\frac{2}{3}{\left(\frac{7}{8}\right)}^r$.

Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.

A marathon is 42.195 kilometres.

In the $k$th training session Rosa will run further than a marathon for the first time.

Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month.

Helen and Jane both commence new jobs each starting on an annual salary of $\$70,000$. At the start of each new year, Helen receives an annual salary increase of $\$2400$.

Let $\$H_n$ represent Helen’s annual salary at the start of her $n$th year of employment.

At the start of each new year, Jane receives an annual salary increase of $3\%$ of her previous year’s annual salary.

Jane’s annual salary, $\$J_n$, at the start of her $n$th year of employment is given by $J_n=70 000{\left(1.03\right)}^{n-1}$.

At the start of year $N$, Jane’s annual salary exceeds Helen’s annual salary for the first time.

An infinite geometric series has first term $u_1=a$ and second term $u_2=\frac{1}{4}{a}^2-3a$, where $a>0$.

Let $f\left(x\right)={\text{e}}^{2\text{sin}\left(\frac{\mathrm{\pi <!--\; \pi \; -->}x}{2}\right)}$, for x > 0.

The k th maximum point on the graph of f has x-coordinate x_k where $k\in <!--\; \in \; -->{\mathbb{Z}}^{+}$.

The first terms of an infinite geometric sequence, $u_n$, are 2, 6, 18, 54, …

The first terms of a second infinite geometric sequence, $v_n$, are 2, −6, 18, −54, …

The terms of a third sequence, $w_n$, are defined as $w_n=u_n+v_n$.

The finite series, $\underset{k=1}{\overset{225}{\sum <!--\; \sum \; -->}}{w}_k$ , can also be written in the form $\underset{k=0}{\overset{m}{\sum <!--\; \sum \; -->}}4r^k$.

The first term of an infinite geometric sequence is 4. The sum of the infinite sequence is 200.

The first two terms of a geometric sequence are $u_1=2.1$ and $u_2=2.226$.

On 1st January 2020, Laurie invests $P in an account that pays a nominal annual interest rate of 5.5 %, compounded quarterly.

The amount of money in Laurie’s account at the end of each year follows a geometric sequence with common ratio, r.

In an arithmetic sequence, $u_1=1.3$ , $u_2=1.4$ and $u_k=31.2$.

Consider the terms, $u_n$, of this sequence such that $n$ ≤ $k$.

Let $F$ be the sum of the terms for which $n$ is not a multiple of 3.

All answers in this question should be given to four significant figures.

In a local weekly lottery, tickets cost $\$2$ each.

In the first week of the lottery, a player will receive $\$D$ for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving $\$10$ is $0.03$. The grand prize in the first week of the lottery is $\$1000$.

If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be $\$2000$ in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.

Gemma and Kaia started working for different companies on January 1st 2011.

Gemma’s starting annual salary was $\$45 000$, and her annual salary increases $2\%$ on January 1st each year after 2011.

Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.