Arithmetic Sequences & Series (+Sigma notation): notes and practice questions

(a) The total cost, $C$, of electricity for a small business varies directly with the number of kilowatt-hours, $E$, consumed.

If the business consumed $1500$ kilowatt-hours of electricity and the total cost was $270$, find a relationship connecting $C$ and $E$.

(b) Find the total cost if the business consumes $2500$ kilowatt-hours of electricity.

(c) Find how many kilowatt-hours were consumed if the total cost was $360.

A new manufacturing plant, 'InnovateTech', began production. On its first day, the plant produced 1200 units. Due to an optimized workflow, the production increased by 80 units on each subsequent day.

Calculate the number of units produced by InnovateTech on day 15 of its operation.

Another plant, 'QuantumFab', also started production on the same day, producing 1000 units. QuantumFab increased its production by 3% of the previous day's output on each subsequent day.

Calculate the number of units produced by QuantumFab on day 15 of its operation.

On which day, $n$, will QuantumFab's production exceed InnovateTech's production for the first time?

(a) A local community group, "Green Spaces", is fundraising for a new park playground. They received an initial donation of £750 and plan to collect £200 each month from local businesses. Their goal is to raise £12,000.

Determine how many more months it will take the group to fundraise for their goal.

(b) To speed up the fundraising, the group decides to increase the monthly contribution by a fixed amount of £$k$ each month, starting from the original £$200$, so they can reach their £$12\,000$ goal within 18 months.

Find the smallest value of $k$ (rounded to the nearest dollar) so that the group can achieve their goal.

(c) If the "Green Spaces" group continues to increase their monthly contributions at the rate found in part (b) and spends none of it, they aim for a larger milestone of £$25\,000$ to include additional facilities.

Determine after how many more months (from the end of the 18-month period in part (b) ) they will have at least £$25\,000$ fundraised.

Tech Innovations is a company that produces a new line of smart devices. In its first year of operation, the company produced 15000 units. Due to efficient scaling, the production increases by 8% of the initial production at the start of each subsequent year.

The annual production figures form an arithmetic sequence.

(a) Calculate the common difference of this sequence.

(b) After $k$ complete years, the annual production will exceed 30000 units for the first time.

Find the value of $k$.

(a) A tech startup's monthly revenue is modelled as a geometric sequence. The revenue in the first month was $R_1$ dollars, and it increases by a constant factor $k$ each month.

(i) Write down an expression in terms of $R_1$ and $k$ for the revenue in the $n$th month, $R_n$.

(ii) Write down an expression for the total revenue generated over the first $N$ months, $S_N$.

(b) The marketing team measures the 'growth potential' $G_n$ of the startup in the $n$th month using the formula $G_n = \log_{10} R_n$.

By writing $G_n$ in terms of $R_1$ and $k$, determine what type of sequence $G_n$ defines, and find an expression for the $n$th term $G_n$. You must clearly show your working and justify your answer.

(c) Find an expression for the sum $T_N$ of the growth potential over the first $N$ months.

(d) Determine whether or not $T_N = \log_{10} S_N$. Justify your answer.

A robotic arm is designed to sweep through a fixed angle of 60 degrees. The arm starts with an initial length of 5 cm and then extends by 2 cm for each subsequent sweep. The path traced by the tip of the arm forms an arc of a circle.

(a) Find the length of the arc traced during the first sweep.

(b) Find the length of the arc traced during the second sweep.

(c) Find the length of the arc traced during the third sweep.

(d) Describe the relationship between the lengths of the arcs traced during the first, second, and third sweeps. Hence, find the lengths of the arcs traced during the fourth and fifth sweeps without performing full calculations.

(e) Determine the area of the sector traced by the arm during its sixth sweep.

A construction company, 'BuildFast', is undertaking a large infrastructure project. The project is divided into 12 distinct phases.

Two teams, Team Alpha and Team Beta, are assigned to complete the project, working in parallel on different aspects.

Team Alpha's efficiency: The first phase takes them $3$ minutes $30$ seconds. Due to a new process, the time Team Alpha takes to complete each subsequent phase is $5$ seconds less than the previous phase.

Team Beta's efficiency: The first phase also takes them $3$ minutes $30$ seconds. Due to a continuous learning curve, the time Team Beta takes to complete each subsequent phase is $0.95$ times the time they took for the previous phase.

(a) (i) State the time Team Alpha takes to complete the third phase of the project.

(a) (ii) Show that Team Beta takes approximately $3$ minutes $10$ seconds (to the nearest second) to complete the third phase of the project.

(b) Both teams complete all $12$ phases of the project. Show that Team Beta completes the entire project faster than Team Alpha.

(c) Hence, state the value of the time difference, correct to the nearest second, between their total project completion times.

A small business is tracking its monthly profit. The profits for the first four months, in thousands of dollars, form an arithmetic sequence. These profits are given by $2a+b$, $a+3b$, $4a-2b$, and $3a+5b-1$ respectively.

Find the value of $a$ and the value of $b$.

Find the least number of months required for the total accumulated profit to exceed $50,000.

(a) TechCraft Innovations starts with an annual production of $1500$ drones. Due to market demand and efficiency improvements, the production increases by $8\%$ each year on the previous year's output.

Justify that the annual drone production over the years forms a geometric sequence, and state its common ratio.

(b) Determine TechCraft Innovations' annual drone production during the third year of operation.

(c) Find the total number of drones produced by TechCraft Innovations during the first $8$ years of operation, correct to the nearest whole number.

(d) After the initial offer, TechCraft Innovations considers an alternative production plan. They want to start with the same annual production of $1500$ drones, but with a fixed annual increase of $d$ drones. The company's financial manager agrees to this new plan if the total production over the $8$ years remains the same as it was in the original geometric plan.

Determine the value of $d$, correct to two decimal places, that satisfies the financial manager's conditions.

(e) Hence, determine the number of years for which the second plan (arithmetic) gives TechCraft Innovations a higher annual production than the first plan (geometric).

A set of numbered cards is created, where each card has a number that is a multiple of $4$. The numbers start from $4$ and go up to $6n$, where $n$ is an even positive integer.

Calculate the total number of cards in the set, in terms of $n$.

Find the probability, in terms of $n$, that a card selected at random from the set shows a number that is divisible by $6$.

A new electronics company, 'TechFlow', launched its flagship product. Let $n$ be the number of years since the product's launch. The sales team recorded the following data for the first two years.

Calculate the percentage increase in units sold from the first year to the second year.

It is assumed that the number of units sold each year will follow a geometric sequence, $u_n$.

Write down the common ratio of the sequence.

Find an expression for $u_n$.

Find the number of units TechFlow expects to sell when $n = 15$. Express your answer to the nearest integer.

In the first year, TechFlow's production facility had a capacity of $5500$ units. The company plans to increase its production capacity by $400$ units every year.

Let $v_n$ represent the production capacity of the facility in year $n$.

Write down an expression for $v_n$.

For the first $12$ years, TechFlow ensures that all units produced are sold. Each unit sold generates a profit of $30.

Calculate the total profit generated from units sold in the first $12$ years.

When $n = k$, the number of units demanded (sales) will, for the first time, exceed the production capacity.

Find $k$.

State whether, for all $n > k$, TechFlow will consistently have sales exceeding its production capacity.

Justify your answer.

A new automated assembly line begins production by manufacturing $15$ components on its first day of operation. Due to continuous optimization, the number of components produced increases by a constant amount each day. On the last day of this initial production phase, the assembly line manufactured $35.4$ components. The daily increase in production is $0.6$ components.

(a) Determine the total number of days this initial production phase lasted.

(b) Calculate the total number of components produced during this initial production phase.

Leo, a baker, starts a new production schedule. On the first day, he bakes $15$ loaves of bread. On each subsequent day, he plans to bake $8$ more loaves than he did the day before.

Calculate the number of loaves Leo would bake on the $10$th day.

Leo plans to bake according to this schedule for a total of $12$ days.

Calculate the total number of loaves he would have baked by the end of the $12$th day.

Leo wants to have at least $1200$ loaves of bread available after $12$ days. He still plans to bake $15$ loaves on the first day, but he will increase the number of loaves baked each subsequent day by $x$ (instead of $8$).

Given that he will still bake for a total of $12$ days, calculate the minimum integer value of $x$ required for him to reach his target.

Leo then decides to experiment with a new type of dough. He drops a small ball of this dough from a height. After the first bounce, the ball reaches a height of $80$ cm. On each successive bounce, the height the ball reaches is $85\%$ of its previous height.

Calculate the height the ball reaches after the $6$th bounce.

Calculate the total vertical distance travelled by the ball during the first $10$ bounces (i.e., the sum of the heights reached from the $1$st bounce to the $10$th bounce).

Calculate the initial height from which Leo dropped the ball before the first recorded bounce.

(a) A start-up company's profit in its first month of operation was $15000. Due to a consistent growth strategy, the profit increased by$750 each subsequent month.

Find the profit for the second and third months.

(b) Write down an expression for $P_n$, the profit in the $n$-th month, in terms of $n$.

(c) The company aims to achieve a monthly profit greater than $35000. Find the smallest number of months,$n$, for which the profit will exceed$35000.

(a) An architect is designing a modular facade for a building. Each module is based on a fundamental rectangular panel, $P$. The vertices of panel $P$ are $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$.

Show that the area of the panel $P$ is $6$ square units.

(b) The design incorporates 30 elements, each obtained by transforming the panel $P$. These elements form a symmetric pattern, with the y-axis acting as a line of symmetry.

The transformation that produces each of the elements on the right side of the design can be represented by a matrix of the form

$M_k = \begin{pmatrix} (1-\frac{k}{15})\cos(k\times12\degree) & -(1-\frac{k}{15})\sin(k\times12\degree) \\ (1-\frac{k}{15})\sin(k\times12\degree) & (1-\frac{k}{15})\cos(k\times12\degree) \end{pmatrix}$

where $k = 0, 1, 2, ..., 14$.

(i) Find the matrix $M_0$. Give your answer in the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a, b, c, d \in \mathbb{Q}$.

(ii) Hence find the coordinates of the image of the vertex $(3,2)$ after it is transformed by the matrix $M_0$.

(c) The matrix $M_k$ can be expressed as the product of a rotation matrix and an enlargement matrix.

(i) Write down, in terms of $k$, the rotation matrix.

(ii) Write down, in terms of $k$, the enlargement matrix.

(iii) Write down, in terms of $k$, the angle of the rotation.

(iv) Write down, in terms of $k$, the scale factor of the enlargement.

(d) Using your answer to part (c)(iv), or otherwise, find the determinant of the matrix $M_k$ in terms of $k$.

(e) Hence, or otherwise, find the total area of the elements in the whole design. Give your answer to three significant figures.

(f) Each element on the left side of the design can be obtained through a transformation of the panel $P$ by applying the matrix $N_k$, where $k = 0, 1, 2, ..., 14$.

Write down the matrix $N_k$ as a product of two matrices.

A mountain climber starts at an altitude of $4500$ meters and descends at a constant rate of $180$ meters per hour. Let $a_n$ represent the climber's altitude after $n$ hours.

Show that the sequence of altitudes $a_n$ forms an arithmetic sequence.

Find the smallest number of hours, $k$, for which the climber's altitude is below $1200$ meters.

Maya is planning for her retirement and decides to invest in two different funds, Fund A and Fund B, both starting with an initial deposit of $12000$ AUD.

For Fund A, Maya makes annual contributions. She contributes $800$ AUD at the end of the first year, $870$ AUD at the end of the second year, $940$ AUD at the end of the third year, and so on. The amount of contribution continues to increase each year, following an arithmetic sequence.

Find the common difference of the annual contributions.

Find the amount of contribution, in AUD, Maya makes at the end of the 10th year.

Show that the total amount of money in Fund A after $n$ years may be expressed as $12000+\frac{n}{2}(1530+70n)$.

Hence or otherwise, find the total amount of money in Fund A at the end of 15 years.

Fund B also starts with an initial deposit of $12000$ AUD. It pays an annual interest rate of $5.5\%$ compounded annually. The amount in Fund B after $n$ years can be expressed as $12000 \times B^n$ where $B\in \mathbb{R}$.

Write down the value of $B$.

Hence or otherwise, show that Fund A will have more money than Fund B at the end of 15 years.

The client is interested in a longer-term investment. Maya finds that it will take at least $m$ complete years for the amount in Fund B to exceed the amount in Fund A.

Find the value of $m$.

Determine the total interest added to Fund B at the end of $m$ years.

Give your answer correct to the nearest dollar.

A productivity coach challenges their client to complete a growing number of tasks each day. On the first day, the client completes $15$ tasks. Each subsequent day, the client completes $3$ more tasks than the previous day.

(a) Write down an expression for $S_n$, the total number of tasks completed after $n$ days.

(b) Hence, find the number of days, $n$, it takes for the client to complete a total of $870$ tasks.

(a) A new social media app, "Connectify", reported a user base of $15.73$ million users at the end of its first year.

(i) Write down $15.73$ million correct to the nearest million.

(ii) Find the percentage error if $15.73$ million is rounded to the nearest million.

Connectify's initial user growth followed an arithmetic progression. In the first month, they gained $5000$ new users. In the second month, they gained $7000$ new users, and in the third month, $9000$ new users.

(b) Find the month during which Connectify gained $35000$ new users.

(c) Calculate the total number of new users Connectify gained in the first $12$ months.

Meanwhile, LinkUp, a rival platform, started with an advertising budget of $100000$ in the first quarter. Due to its success, they decided to increase their budget by $15\%$ each subsequent quarter.

(d) Determine the first quarter in which LinkUp's advertising budget exceeds $500000.

(e) Find the first quarter that the total advertising expenditure by LinkUp, since the start, exceeds $2000000.

The daily production, $P_n$, of a new electronic gadget on the $n$-th day of operation is modelled by the arithmetic sequence formula $P_n = 138 + 12n$.

(a) State the production on the first day.

(b) Calculate the production on the $30$th day.

(c) Explain why a production target of $500$ gadgets cannot be met exactly on any particular day according to this model.