Geometric Sequences & Series: notes and practice questions

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 water purification system reduces the impurity concentration in water. Initially, the impurity concentration is 320 mg/L. Each filter stage removes 15% of the impurities present before that stage.

Show that the impurity concentration after the fourth filter stage is approximately 167 mg/L, to the nearest mg/L.

(b) Find the minimum number of filter stages required for the impurity concentration to be less than 25 mg/L.

(c) A technician records the impurity concentration after each stage. Calculate the sum of the concentrations recorded after the first, second, third, and fourth stages. Give your answer to one decimal place.

A new type of eco-friendly battery is being tested for its long-term performance. In its first full charge cycle, the battery delivers a total capacity of 2500 mAh. Due to gradual degradation, the battery's maximum capacity for each subsequent charge cycle is only 92% of the capacity delivered in the previous cycle.

(a) Calculate the total cumulative charge capacity delivered by the battery over its first 8 full charge cycles.

(b) The manufacturer guarantees that the battery will deliver a total cumulative charge capacity of at least 31500 mAh over its entire lifespan. Determine if this battery meets the manufacturer's guarantee.

(c) Suggest a limitation of the given model for battery degradation.

A new manufacturing plant starts production on 1 January 2024. In January 2024, the plant produces 200 units. The production is expected to increase each month, modelled by a geometric sequence with a common ratio of 1.08.

(a) Calculate the number of units the plant is expected to produce in October 2024.

(b) Calculate the total number of units the plant is expected to produce

(i) in the year 2024.

(ii) in the year 2025.

A new online challenge is introduced. In the first hour, 120 people participate. Due to decreasing novelty, the number of new participants joining each subsequent hour is 82% of the number of new participants who joined the previous hour.

(a) Show that the number of new participants joining in the 7th hour is 36, to the nearest whole number.

(b) Find the number of hours for which the number of new participants joining in that hour is greater than 15.

(c) Find the total number of people who have heard about the challenge by the end of the 5th hour, to the nearest whole number.

(a) A population of bacteria in a petri dish grows according to a geometric sequence. On day 2, there are $12$ bacteria. On day 5, there are $324$ bacteria.

Find the initial number of bacteria (first term, $a$) and the daily growth factor (common ratio, $r$).

(b) Find:

(i) the number of bacteria on day 8.

(ii) the total number of bacteria observed from day 1 to day 8.

(c) Find the first day $n$ when the number of bacteria exceeds $500 \, 000$.

A specialized bouncy ball is dropped from a certain height. After the first bounce, it reaches a maximum height of $u_1$ meters. After each subsequent bounce, the maximum height it reaches is a constant fraction of the previous maximum height.

The first bounce reaches a height of 3 meters. The sum of all the maximum heights reached after each bounce (i.e., $u_1 + u_2 + u_3 + \dots$) is 15 meters.

Find the common ratio, $r$, for the sequence of maximum bounce heights.

Find the least value of $n$ such that the maximum height reached after the $n$-th bounce, $u_n$, is less than 0.1 meters.

(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 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 new productivity app is launched, and its user base grows virally. The number of active users triples every week. There were 150 active users at the end of Week 1.

The following table shows the estimated number of active users, $u_n$, for this app.

The values of $u_n$ form the terms of a sequence.

(a) Complete the table by adding the two missing values.

(b) (i) Find the value of $n$ when the number of active users reaches 2,952,450.

(ii) Hence or otherwise, find the total number of new users generated from the start of the campaign until the end of week $n$ (from part (b)(i) ). Give your answer correct to the nearest thousand users.

(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 specialized bouncy ball is dropped from a height of 16 meters. After the first bounce, it reaches a height of 8 meters. The height reached after each subsequent bounce is half of the height reached after the previous bounce.

(a) Calculate the height the ball reaches after its 4th bounce.

(b) Find the total vertical distance the ball has traveled upwards after 6 bounces.

(c) Determine the total vertical distance the ball travels from the moment it is dropped until it theoretically comes to rest.

A new mobile application, 'MindFlow', launched on 1 January 2024. In January 2024, it had 500 active users. The number of active users is expected to grow each month, following a geometric sequence with a common ratio $r = 1.08$.

(a) Calculate the number of active users 'MindFlow' is expected to have in December 2024.

(b) The developers want to track the total user base.

(i) Calculate the total number of active users 'MindFlow' is expected to accumulate during the entire year 2024.

(ii) Calculate the total number of active users 'MindFlow' is expected to accumulate during the year 2025.

A digital chain letter campaign begins with a small group of enthusiasts. In the first round of distribution, 150 unique letters were sent. Each recipient is instructed to forward the letter to 3 new individuals in the next round. The rounds of distribution occur daily.

The number of unique letters distributed in each round, $u_n$, forms a geometric sequence.

Complete the table by adding the two missing values.

Find the value of $n$ when the number of unique letters distributed in a single round is 109350.

Hence or otherwise, find the total number of new letters distributed across all rounds as the number of letters in a round increases from 150 to 109350. Give your answer correct to the nearest thousand new letters.

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 scientist is observing the growth of a rare crystalline structure in a controlled environment. In the first hour ($r=1$), $12$ grams of the crystal formed. Each subsequent hour, the mass of new crystal formed is three times the mass formed in the previous hour.

(a) Find an expression for the total mass of crystal formed after $n$ hours, in its most simplified form.

(b) Find an expression for the total mass of crystal formed after $2n$ hours, in its most simplified form.

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 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.