Show that one of the real roots is equal to 1.

Calculate, in CAD, the total amount John pays for the bicycle.

Write down the common ratio of the sequence.

The following diagram shows [CD], with length $b\text{ cm}$, where $b>1$. Squares with side lengths $k\text{ cm},k^2\text{ cm},k^3\text{ cm},\dots$, where , are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.

The total sum of the areas of all the squares is $\frac{9}{16}$. Find the value of $b$.

Show that $abc=8$.

Find the sum of the first 8 terms.

Find the values of θ which give the greatest value of the sum.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Hence or otherwise, show that the series is convergent.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Show that, at these times, $\tan 6t=2$.

Find the value of $N$.

Given that $J_n$ follows a geometric sequence, state the value of the common ratio, $r$.

Find Jane’s total earnings at the start of her $10$th year of employment. Give your answer correct to the nearest dollar.

Find the maximum displacement of $P$, in metres, from its initial position.

Find an expression for r in terms of θ.

Find the common ratio.

Show that $\text{P}(A\cup B)=\text{P}(A)+\text{P}(A^{\prime }\cap B)$.

Find the least value of $n$ such that $S_n>5543$.

Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of $n$ for the value of the grand prize in the $n\text{th}$ week of the lottery.

Find an expression for $s$ in terms of $t$.

Find the value of $r$, the common ratio of the sequence.

The first term in an arithmetic sequence is $4$ and the fifth term is ${\log }_2 625$.

Find the common difference of the sequence, expressing your answer in the form ${\log }_2 p$, where $p\in \mathrm{\mathbb{Q}}$.

Show that $H_n=2400n+67 600$.

Find the sum to infinity of this sequence.

Given that $q=8d^2$, find the other two real roots.

For the value of $N$ found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Write down an expression for $f^{-1}\left(x\right)$.

Find Gemma’s annual salary for the year 2021, to the nearest dollar.

Find the smallest value of $n$ for which $u_n$ is less than ${10}^{-3}$.

Find the value of $k$.

Find the total depth that the post has been driven into the ground after 10 strikes of the hammer.

Calculate the café’s total profit for the first 12 weeks.

Find the first term of the sequence, $u_1$.

Find $S_{\infty }$.

Find the value of $u_{10}$.

Find the value of $f^{-1}\left(\sqrt{32}\right)$.

A geometric sequence has $u_4=-70$ and $u_7=8.75$. Find the second term of the sequence.

Find the least value of n for which S_n > 163.

Consider a geometric sequence where the first term is 768 and the second term is 576.

Find the least value of $n$ such that the $n$th term of the sequence is less than 7.

Find the times when $P$ comes to instantaneous rest.

Find the value of $r$.

Hence show that $\frac{v_2}{v_1}=\frac{v_3}{v_2}=-{\text{e}}^{-\frac{\mathrm{\pi }}{2}}$.

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

Find the common ratio of the sequence.

Find the volume of the smallest slice of pie.

Find the distance that the post is driven into the ground by the eighth strike of the hammer.

Calculate the total distance Carlos runs in the first year.

Find the value of $u_5$.

The apple pie has a volume of $61 425 {\text{cm}}^3$.

Find the total number of slices Mia can cut from this pie.

Find the sum of the first 30 terms of the sequence.

Find the value of the common ratio for this sequence.

John purchased the bicycle in 2008.

Justify why John should not insure his bicycle in 2019.

Find the value of $n$ for which $u_n=2$.

Show that $a=8$.

Find the value of $p$ and the value of $q$.

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

Find the least value of $n$ such that $S_{\infty }-S_n<0.001$.

Find the value of the bicycle during the 5th year. Give your answer to two decimal places.

Calculate, in years, when the bicycle value will be less than 50 USD.

Find the total amount John has paid to insure his bicycle for the first 5 years.

Write down the distance Rosa runs in the third training session;

Write down the distance Rosa runs in the $n$th training session.

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

Find the distance Carlos runs in the fifth month of training.

Calculate the tea-shop’s total profit for the first 12 weeks.