Show that $p=\frac{2}{3}$.

Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

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

Calculate the number of positive terms in the sequence.

At point P, the graphs of $f$ and $g$ intersect for the 21st time. Find the coordinates of P.

Find u₈.

By using a suitable table of values or otherwise, determine the smallest positive integer, greater than $1$, that is both a triangular number and a pentagonal number.

Hence find the value of n such that $\sum _{k=1}^n{x}_k=861$.

Given that x_{k + 1} = x_k + a, find a.

the value of $r$;

Find the common difference.

Find the common difference.

Hence show that $P_5\left(n\right)=\frac{n\left(3n-1\right)}{2}$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

Show that $p=\frac{2}{3}$.

The sum of the first $n$ terms of the series is $\ln \left(\frac{1}{x^3}\right)$.

Find the value of $n$.

Given that the $n$^th term of this sequence is $-33$, find the value of $n$.

Find the common difference, $d$.

Find the approximate number of fish in the lake at the start of 2042.

Find the tenth term.

Find the sum of the first $20$ terms.

Find the value of $m$.

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.

Show that the tank will never be completely filled using this pump.

find the value of $d$.

Consider an arithmetic sequence where $u_8=S_8=8$. Find the value of the first term, $u_1$, and the value of the common difference, $d$.

Find the common difference.

Find the tenth term.

Show that the volume of the tank is $624 000 {\text{m}}^3$, correct to three significant figures.

The following diagram shows part of the graph of $g$ reflected in the $x$-axis. It also shows part of the graph of $f$ and the point P.

Find an expression for the area of the shaded region. Do not calculate the value of the expression.

Find the two smallest non-zero values of $x$ for which $f(x)=g(x)$.

Find the exact value of $S_k$.

Show that $F=3240$.

Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number.

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

On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.

Show that there will be approximately 2645 fish in the lake at the start of 2020.

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.

Find the first term.

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_n$.

Show that $u_1=4$.

Find the common ratio of this sequence.

State the value of the first term, $u_1$.

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

Find the value of $k$.

For that day find the weight of Sergei’s first lift.

Write down the common difference, $d$.

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

Find the value of $r$.

Write down the first three non-zero terms of $w_n$.

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

determine the value of $\sum _{r=1}^N{u}_r$.

An infinite geometric series is given as $S_{\mathrm{\infty }}=a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\dots$, $a\in {\mathbb{Z}}^{+}$.

Find the largest value of $a$ such that .

Find the year in which the comet was seen from Earth for the fifth time.

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

Find the distance from the base of this ladder to the top rung.

the value of $d$;

For that day find how much weight was added after each lift.

Determine how many times the comet has been seen from Earth up to the year 2014.

Diagram $n$ is formed with 52 sticks. Find the value of $n$.

Calculate the total distance Carlos runs in the first year.

Write down the number of hours that the pump was pumping fuel into the tank.

The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.

Find the amount of fuel pumped into the tank in the $\text{13th}$ hour.

Find the total amount of fuel pumped into the tank in the first $8$ hours.

John purchased the bicycle in 2008.

Justify why John should not insure his bicycle in 2019.

Find the total number of sticks used by Tomás for all 24 diagrams.

Find the value of $n$ such that $u_n=0$.

Find $h$, the height of the tank.

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 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.