Sergei is training to be a weightlifter. Each day he trains at the local gym by lifting a metal bar that has heavy weights attached. He carries out successive lifts. After each lift, the same amount of weight is added to the bar to increase the weight to be lifted.

The weights of each of Sergei’s lifts form an arithmetic sequence.

Sergei’s friend, Yuri, records the weight of each lift. Unfortunately, last Monday, Yuri misplaced all but two of the recordings of Sergei’s lifts.

On that day, Sergei lifted 21 kg on the third lift and 46 kg on the eighth lift.

The Osaka Tigers basketball team play in a multilevel stadium.

The most expensive tickets are in the first row. The ticket price, in Yen (¥), for each row forms an arithmetic sequence. Prices for the first three rows are shown in the following table.

For a study, a researcher collected 200 leaves from oak trees. After measuring the lengths of the leaves, in cm, she produced the following cumulative frequency graph.

Mia baked a very large apple pie that she cuts into slices to share with her friends. The smallest slice is cut first. The volume of each successive slice of pie forms a geometric sequence.

The second smallest slice has a volume of $30 {\text{cm}}^3$. The fifth smallest slice has a volume of $240 {\text{cm}}^3$.

The $pH$ of a solution is given by the formula $pH=-{\log }_{10} C$ where $C$ is the hydrogen ion concentration in a solution, measured in moles per litre (${\text{Ml}}^{-1}$).

A ball is dropped from a height of $1.8$ metres and bounces on the ground. The maximum height reached by the ball, after each bounce, is $85\%$ of the previous maximum height.

Juliana plans to invest money for $10$ years in an account paying $3.5\%$ interest, compounded annually. She expects the annual inflation rate to be $2\%$ per year throughout the $10$-year period.

Juliana would like her investment to be worth a real value of $\$4000$, compared to current values, at the end of the $10$-year period. She is considering two options.

Option 1: Make a one-time investment at the start of the $10$-year period.

Option 2: Invest $\$1000$ at the start of the $10$-year period and then invest $\$x$ into the account 
                at the end of each year (including the first and last years).

The ticket prices for a concert are shown in the following table.

• A total of $600$ tickets were sold.
• The total amount of money from ticket sales was $\$7816$.
• There were twice as many adult tickets sold as child tickets.

Let the number of adult tickets sold be $x$, the number of child tickets sold be $y$, and the number of student tickets sold be $z$.

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$, which have a magnitude of at least $M$. For a particular region the equation is

${\log }_{10} N=a-M$, for some $a\in \mathrm{ℝ}$.

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$.

The equation for this region can also be written as $N=\frac{b}{{10}^M}$.

The expected length of time, in years, between earthquakes with a magnitude of at least $M$ is $\frac{1}{N}$.

Within this region the most severe earthquake recorded had a magnitude of $7.2$.

In an arithmetic sequence, the first term is 8 and the second term is 5.

The pH of a solution measures its acidity and can be determined using the formula pH $=-{\log }_{10} C$, where $C$ is the concentration of hydronium ions in the solution, measured in moles per litre. A lower pH indicates a more acidic solution.

The concentration of hydronium ions in a particular type of coffee is $1.3\times 10{}^{-5}$ moles per litre.

A different, unknown, liquid has $10$ times the concentration of hydronium ions of the coffee in part (a).

In this question, give all answers correct to 2 decimal places.

Raul and Rosy want to buy a new house and they need a loan of $170 000$ Australian dollars ($\mathrm{AUD}$) from a bank. The loan is for $30$ years and the annual interest rate for the loan is $3.8\%$, compounded monthly. They will pay the loan in fixed monthly instalments at the end of each month.

Tomás is playing with sticks and he forms the first three diagrams of a pattern. These diagrams are shown below.

Tomás continues forming diagrams following this pattern.

Tomás forms a total of 24 diagrams.

Consider the geometric sequence $u_1=18,u_2=9,u_3=4.5,\dots <!--\; \dots \; -->$.

The following table shows four different sets of numbers: $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$.

The first three terms of a geometric sequence are $\ln <!--\; \; -->x^{16}$, $\ln <!--\; \; -->x^8$, $\ln <!--\; \; -->x^4$, for $x>0$.

Siân invests $50000$ Australian dollars (AUD) into a savings account which pays a nominal annual interest rate of $5.6$ % compounded monthly.

The first two terms of an infinite geometric sequence, in order, are

$2{\log }_{2}x,{\log }_{2}x$, where $x>0$.

The first three terms of an arithmetic sequence, in order, are

${\log }_{2}x,{\log }_2\left(\frac{x}{2}\right),{\log }_2\left(\frac{x}{4}\right)$, where $x>0$.

Let $S_{12}$ be the sum of the first 12 terms of the arithmetic sequence.

Give your answers in this question correct to the nearest whole number.

Imon invested $25 000$ Singapore dollars ($\text{SGD}$) in a fixed deposit account with a nominal annual interest rate of $3.6\%$, compounded monthly.

The company Snakezen’s Ladders makes ladders of different lengths. All the ladders that the company makes have the same design such that:

the first rung is 30 cm from the base of the ladder,

the second rung is 57 cm from the base of the ladder,

the distance between the first and second rung is equal to the distance between all adjacent rungs on the ladder.

The ladder in the diagram was made by this company and has eleven equally spaced rungs.

In an arithmetic sequence, u₁ = −5 and d = 3.

Yejin plans to retire at age 60. She wants to create an annuity fund, which will pay her a monthly allowance of $4000 during her retirement. She wants to save enough money so that the payments last for 30 years. A financial advisor has told her that she can expect to earn 5% interest on her funds, compounded annually.

A geometric sequence has a first term of $\frac{8}{3}$ and a fourth term of $9$.

A disc is divided into $9$ sectors, number $1$ to $9$. The angles at the centre of each of the sectors $u_n$ form an arithmetic sequence, with $u_1$ being the largest angle.

It is given that $u_9=\frac{1}{3}{u}_1$.

Iron in the asteroid 16 Psyche is said to be valued at $8973$ quadrillion euros $\left(\text{EUR}\right)$, where one quadrillion $={10}^{15}$.

James believes the asteroid is approximately spherical with radius $113 \text{km}$. He uses this information to estimate its volume.

Give your answers to this question correct to two decimal places.

Gen invests $2400 in a savings account that pays interest at a rate of 4% per year, compounded annually. She leaves the money in her account for 10 years, and she does not invest or withdraw any money during this time.

The speed of light is $\text{300}\text{000}$ kilometres per second. The average distance from the Sun to the Earth is 149.6 million km.

The intensity level of sound, $L$ measured in decibels (dB), is a function of the sound intensity, $S$ watts per square metre (W m⁻²). The intensity level is given by the following formula.

$L=10\text{lo}{\text{g}}_{10}\left(S\times <!--\; \times \; -->{10}^{12}\right)$, $S$ ≥ 0.

Sophia pays $\$200$ into a bank account at the end of each month. The annual interest paid on money in the account is $3.1\%$ which is compounded monthly.

The average rate of inflation per year over the $5$ years was $2\%$.

An arithmetic sequence has $u_1=\text{lo}{\text{g}}_c\left(p\right)$ and $u_2=\text{lo}{\text{g}}_c\left(pq\right)$, where $c>1$ and $p,q>0$.

Tommaso and Pietro have each been given $1500$ euro to save for college.

Pietro invests his money in an account that pays a nominal annual interest rate of $2.75\%$, compounded half-yearly.

Charlie and Daniella each began a fitness programme. On day one, they both ran $500 \text{m}$. On each subsequent day, Charlie ran $100 \text{m}$ more than the previous day whereas Daniella increased her distance by $2\%$ of the distance ran on the previous day.

Calculate how far

Katya approximates $\mathrm{\pi }$, correct to four decimal places, by using the following expression.

$3+\frac{1}{6+{\displaystyle \frac{13}{16}}}$

Let $p=\frac{\cos <!--\; \; -->x+\sin <!--\; \; -->y}{\sqrt{w^2-<!--\; -\; -->z}}$,

where $x={36}^{\circ <!--\; \circ \; -->},y={18}^{\circ <!--\; \circ \; -->},w=29$ and $z=21.8$.

Inspectors are investigating the carbon dioxide emissions of a power plant. Let $R$ be the rate, in tonnes per hour, at which carbon dioxide is being emitted and $t$ be the time in hours since the inspection began.

When $R$ is plotted against $t$, the total amount of carbon dioxide produced is represented by the area between the graph and the horizontal $t$-axis.

The rate, $R$, is measured over the course of two hours. The results are shown in the following table.