Exercises: Geometric progressions

Analyse the following numerical sequences. Your task is to identify which of them are geometric progressions. For those that are, state their common ratio ($r$).

1. a) $2, 6, 18, 54, \dots$
2. b) $5, 10, 15, 20, \dots$
3. c) $80, 40, 20, 10, \dots$
4. d) $3, -3, 3, -3, \dots$

Look at the following figure. We start with a large square with side $8 \text{ cm}$. The next square is formed with a side length that is half the previous one, and so on.

Calculate the side length of the first 4 squares and write the general term $a_n$ of the sequence formed by these lengths.

Imagine a viral video on social media gets shared very quickly. One person uploads the video (minute 0). At minute 1, 5 new people watch it. At minute 2, each of those 5 people shares it with 5 other new people, and so on.

1. Write down the first four terms of the sequence of new viewers per minute.

2. What is the common ratio $r$ of growth?

3. Write the formula for the general term $a_n$.

We have a sheet of paper with an initial thickness of $0.1 \text{ mm}$.

If we fold the sheet in half, the thickness doubles. If we fold it again, it doubles again.

Consider $a_1$ as the thickness after the first fold.

1. Calculate the thickness of the sheet after 1, 2, 3 and 4 folds.

2. What thickness will the stack of paper have after 10 folds? Express the result in millimetres.

A rubber ball is dropped from a certain height. On each bounce, the ball reaches a height that is half of the previous height.

If on the first bounce ($a_1$) it reaches a height of $160 \text{ cm}$:

1. a) Write down the sequence of heights for the first 5 bounces.
2. b) What is the common ratio $r$ of this progression?
3. c) What height will it reach on the eighth bounce?

Given a geometric progression where the first term is $a_1 = 3$ and the common ratio is $r = 2$:

1. a) Calculate the value of the term in position 7 ($a_7$).
2. b) Calculate the value of the term in position 10 ($a_{10}$).

Use the formula for the general term: $a_n = a_1 \times r^{n-1}$.

A grandfather decides to save up for his grandson following a curious mathematical rule. In the first month he puts 2 euros into a piggy bank. In the second month he puts 6 euros. In the third month he puts 18 euros.

If he continues with this logic:

1. Check that it is a geometric progression and find the common ratio $r$.

2. Calculate how much money he will have to put into the piggy bank in the fifth month.

We have a geometric progression formed by fractions. We know that the first terms are:

1. a) Identify the first term $a_1$ and the common ratio $r$.
2. b) Calculate the term $a_6$ without needing to multiply step by step, using the general formula.

David is solving a classroom exercise: "Find the sixth term of a geometric progression whose first term is $a_1 = 6$ and whose common ratio is $r = 3$". In his notebook he has written the following:

1. a) Identify and explain clearly the conceptual and order of operations error that David has made.
2. b) Write the formula for the general term correctly and calculate the actual value of $a_6$.
3. c) If David had applied his flawed procedure to calculate only the second term ($a_2$), what result would he have obtained compared to the actual value? Explain why David's mistake causes the error to skyrocket dramatically as the index $n$ increases.

Design a geometric progression that simultaneously meets these three conditions:

1. Its terms alternate in sign: the first is positive, the second negative, the third positive, etc.

2. The third term is exactly $a_3 = 12$.

3. The fifth term is exactly $a_5 = 48$.

1. a) State the mathematical relationship between $a_3$ and $a_5$ using the common ratio $r$, deduce the two possible numerical values of $r$, and justify which one you must choose to satisfy the alternating sign condition.
2. b) From the value of the common ratio, determine the first term $a_1$ with reasoning.
3. c) Write the first four terms of your geometric progression and explain why, in a progression with a negative common ratio and a positive first term, it is impossible for a term with an odd index to be negative.

A video by a famous streamer has gone viral. In the first hour it had 5 views. Due to the platform's algorithm, with each passing hour, the number of new views is multiplied by 3 compared to the previous hour.

1. a) Write down the first five terms of the sequence representing the new views per hour.
2. b) What is the common ratio $r$ of this geometric progression?
3. c) Write the general term $a_n$ that allows calculating the views in hour $n$.
4. d) How many new views will the video have in hour 8 exactly?

In a role-playing game, the experience required to level up follows a geometric progression. We know that to reach level 2 you need 150 points of experience, and to reach level 5 you need 1 200 points.

Calculate:

1. a) The common ratio $r$ of the progression.
2. b) The experience required for the first level ($a_1$).
3. c) The experience required to reach level 10.

Look at the following fractal figure. We start with an equilateral triangle of area $64\text{ cm}^2$ (Stage 1). At each subsequent step, in each of the triangles that remain unpainted, we paint the central triangle formed by the midpoints of its sides (inverted triangle), which has an area equal to $1/4$ of the triangle containing it.

Considering the sequence formed by the total area of the new triangles coloured at each stage (starting from Stage 2):

1. a) Write down the first three terms of the sequence.
2. b) What is the common ratio of this progression?
3. c) Calculate the sum of the areas of all the coloured triangles if we were to repeat the process infinitely many times.

Cian wants to buy an electric scooter that costs 400 €. He decides to save up by following a strict plan: in the first week he saves 2 €, in the second week 4 €, in the third 8 €, and so on, doubling the amount each week.

1. a) How much money will he have to save in week 8?
2. b) How much money will he have saved in total by the end of week 8?
3. c) Will he have saved enough money to buy the scooter by the end of week 8? If not, how many weeks does he need in total?

A rubber ball is dropped from an initial height of 20 meters. On each bounce, the ball loses energy and only reaches 80% of the previous height ($r = 0.8$).

1. a) Calculate the height the ball reaches after the third bounce.
2. b) Calculate the total distance traveled by the ball (downward and upward) at the moment it touches the ground for the fourth time.

Note: Keep in mind that the first fall is 20 m, but afterwards it goes up and down.

We have a square of side length 10 cm. We join the midpoints of its sides to form a second inscribed square. We repeat the process with the second square to form a third, and so on.

1. a) Knowing that the area of each new square is half of the previous one, write down the sequence of the areas of the first 5 squares.
2. b) Is it an arithmetic or geometric progression? Identify the first term and the common difference or common ratio.
3. c) Calculate the sum of the areas of the first 10 generated squares.

A biologist is studying a colony of bacteria in a Petri dish. Initially, he counts 500 bacteria. He observes that the population increases by 20% every hour.

1. a) Explain why this growth corresponds to a geometric progression and determine the common ratio $r$.
2. b) Write the general term $a_n$ that describes the number of bacteria after $n$ hours. (Consider $a_0 = 500$ as the start, or adjust the formula for $a_1$ as the amount after 1 hour).
3. c) Approximately how many bacteria will there be after 12 hours?

Calculate the product of the first 6 terms of a geometric progression, knowing that the first term is $a_1 = 3$ and the fourth term is $a_4 = 24$.

In the Science laboratory of a secondary school in Galway, an experiment is carried out by dropping a ball from a height of $3\text{ m}$. After each bounce off the ground, the ball rebounds reaching a maximum height equivalent to $60\,\%$ of the height reached in the previous bounce.

1. a) Estimate before calculating: If the ball were to make infinitely many ideal bounces, do you think the sum of all the heights reached in the bounces ($a_1 + a_2 + a_3 + \ldots$) will exceed $10\text{ meters}$ or will it be less than $6\text{ meters}$? Briefly justify your initial intuition without using formulas.
2. b) Model the situation: find the height reached on the first bounce ($a_1$) and write the expression for the general term $a_n$ of the height corresponding to bounce number $n$.
3. c) Apply the formula for the sum of infinitely many terms ($S_\infty$) to obtain the theoretical total sum of the heights of all the bounces.
4. d) Compare the exact value obtained with your initial estimate. How would you explain to a classmate in your own words that the sum of infinitely many positive quantities can result in an exact, finite number?

According to demographic challenge data from the Central Statistics Office (CSO), a municipality in County Leitrim had $1,250$ registered inhabitants as of January 1, 2018. Due to population aging and emigration, its population is observed to decline at a constant rate of $5\,\%$ per year.

1. a) Model the evolution of the population using a geometric progression: calculate the common ratio $r$, state the value of the initial term $a_1$ (population in 2018), and formulate the general term $a_n$, where $n=1$ corresponds to the year 2018, $n=2$ to 2019, and so on.
2. b) Calculate the projected population for the year 2023 ($n = 6$), rounding to the nearest whole number of people.
3. c) The Health Service Executive (HSE) establishes that if the registered population falls below $900$ inhabitants, the local medical clinic loses daily in-person care. Check numerically whether by the year 2025 ($n = 8$) the municipality will have already crossed that critical threshold or if it will manage to maintain the service.

A biologist is studying a very aggressive strain of bacteria in a Petri dish. Initially, he counts 500 bacteria. He observes that the population increases by 15% every hour.

For the experiment to be controllable, the population must not exceed 2 000 000 bacteria. Using a calculator and your knowledge of the general term of a geometric progression:

1. Write the expression for the general term $a_n$ that describes the number of bacteria as a function of the elapsed hours $n$.

2. Calculate how many bacteria there will be exactly after 24 hours.

3. Determine the maximum number of full hours that can pass before the culture exceeds the safety limit of 2 million.

A special rubber ball is dropped from a height of 2 metres. Due to energy loss, each time it bounces off the ground, it only rises to 80% of the height from which it previously fell.

This behaviour generates a sequence of maximum heights reached after each bounce.

1. Calculate the height reached by the ball after the fifth bounce.

2. Imagine we allow the ball to bounce indefinitely until it stops. Calculate the total distance (downwards and upwards) that the ball will have travelled from when it was released until it stops completely. Be careful: the first drop occurs only once, but the bounces involve going up and down the same distance.

A famous youtuber launches a charity campaign. On the first day, they donate 10 euros. They commit that each subsequent day they will donate an amount such that the sum of the donation on that day and the previous day is three times what they donated the previous day.

Note: Read the problem carefully, it is a math puzzle about the common ratio $r$.

1. Calculate how much they donate on the second and third days.

2. Prove that the amounts donated day by day form a geometric progression and find its common ratio $r$.

3. How much money will they have donated in total after one week (7 days)?

We are designing an artistic logo based on a sequence of inscribed squares.

We start with an outer square of side length $L_1 = 20 \text{ cm}$.

The second square is formed by joining the midpoints of the sides of the first square. The third square joins the midpoints of the second, and so on.

1. Use Pythagoras' Theorem to find the side length of the second square ($L_2$) and the third square ($L_3$).

2. Determine the common ratio of the geometric progression formed by the areas of these squares.

3. If we could draw infinitely many squares inwards following this pattern, what would be the sum of the areas of all the squares (the total shaded area)?

A technology company offers two types of salary contract for a programmer position over 5 years:

• Contract A: Starting salary of 24 000 € per year, with a fixed increase of 2 000 € each year (Arithmetic Progression).

• Contract B: Starting salary of 22 000 € per year, with an 8% increase each year relative to the previous year (Geometric Progression).

• Write the nth term $a_n$ for Contract A and $b_n$ for Contract B.

• Calculate the annual salary in the fifth year for both contracts. Which one pays more in that specific year?

• Calculate the total amount of money earned over the 5 years with each contract ($S_5$). Which contract is more beneficial in the long term (in terms of cumulative total)?

We have a geometric progression for which we do not know the common ratio $r$ or the first term $a_1$. However, we know that the third term is 12 ($a_3 = 12$) and the sixth term is 96 ($a_6 = 96$).

1. Set up a system of equations or use the nth term formula to find the common ratio $r$.

2. Once you have the common ratio, calculate the first term $a_1$.

3. Calculate the product of the first 6 terms of the sequence ($P_6$).

The Sierpinski Triangle is a famous fractal that is built from an equilateral triangle.

The process is as follows (in the figure, the "black" triangles are drawn in green):

• Step 0: We have a black equilateral triangle with an area of $80 \text{ cm}^2$.

• Step 1: We cut out (remove) the inverted central triangle, whose area is $1/4$ of the previous triangle. We are left with 3 smaller black triangles.

• Step 2: From each of the 3 remaining black triangles, we again cut out the central triangle (which is $1/4$ of the area of its containing triangle).

1. Write the sequence that represents the total black area that remains in the figure at each step ($A_0, A_1, A_2, ...$).

2. What is the ratio $r$ of this decreasing geometric progression?

3. Calculate how much black area will remain after applying the process 5 times (Step 5).

We want to interpolate 4 numbers between 3 and 96 to form a geometric progression of 6 terms in total. That is, we aim to construct the sequence:

1. Identify the position occupied by 3 and 96 in the progression ($a_1$ and $a_n$).

2. Use the nth term formula to rearrange and calculate the common ratio $r$.

3. Write down the four numbers we have interpolated.

4. If we wanted to sum all 6 of these numbers, what would the result be?

The mathematics department poses the following challenge: "In a geometric progression, it is known that the fourth term is $a_4 = 24$ and the seventh term is $a_7 = 192$. Calculate the sum of the first 7 terms ($S_7$)".

Two students in the class propose different strategies:

• Aoife proposes using the direct relationship $a_7 = a_4 \times r^{7-4}$ to find the common ratio $r$, going back from $a_4$ by dividing by $r^3$ to find $a_1$, and finally calculating the sum with the closed formula $S_7 = \frac{a_1(r^7 - 1)}{r - 1}$.

• Cian proposes setting up a system of two equations with two unknowns: $\begin{cases} 24 = a_1 \times r^3 \\ 192 = a_1 \times r^6 \end{cases}$, solving it by solving for $a_1$ in the first and substituting into the second, and then calculating the 7 terms ($a_1, a_2, \ldots, a_7$) one by one by hand to add them up directly, avoiding using the sum formula.

1. a) Solve the problem meticulously applying both methods and check whether both lead to the same numerical result.
2. b) Compare the two approaches: what are the weaknesses of Cian's procedure compared to Aoife's strategy in terms of efficiency and risk of error?
3. c) Imagine that the data were $a_{10} = 50$, $a_{30} = 51{,}200$, and the sum of the first 40 terms ($S_{40}$) was requested. Justify with reasoning why Cian's method becomes unfeasible in practice and which of the two approaches demonstrates a mature algebraic understanding.

An entrepreneur from Cork is looking for funding to kick off her project during the first 6 months. Two business incubators offer her non-repayable monthly grant proposals:

• Plan A (Accelerate Programme): She receives $€600$ the first month, and each subsequent month the amount increases by a fixed amount of $€100$ compared to the previous one ($€600, €700, €800, \dots$).

• Plan B (Impulse Fund): She receives $€300$ the first month, and each subsequent month the amount increases by $30\,\%$ compared to that of the previous month ($r = 1.30$).

1. a) Identify what type of progression (arithmetic or geometric) describes each of the options and write the mathematical expression of the general term of the monthly amount received in month $n$ for both plans.
2. b) Calculate the amount she would receive month by month in each plan during the 6 months of the program. In which specific month does the individual monthly payment of Plan B exceed that of Plan A for the first time?
3. c) Use the corresponding summation formulas to calculate the total accumulated financial amount after the full 6 months with each of the two plans. In view of the result, which of the two plans provides her with more money in total, and why does geometric growth not always turn out to be the most advantageous option over short periods, even though in the long run it is exponentially superior?

Topic challenge. Constant savings or multiplier? Design a one-year savings plan to finance a personal project.

1. Phase 1: Real-world data. Go to the CSO website (or the Central Bank of Ireland) and look up the latest published year-on-year inflation rate (CPI, e.g., $2.8\,\%$). Set an initial savings amount for the first month ($a_1 = 20$ €).

2. Phase 2: Modeling. Consider two options for $12$ months:

3. Plan A (Arithmetic progression): You increase your monthly contribution by a fixed difference of $d = 3$ €. Find the general term $a_n$ and the accumulated capital using the sum $S_{12}$.

4. Plan B (Geometric progression): You increase your contribution by $10\,\%$ each month compared to the previous month ($r = 1.10$). Deduce the formula for $b_n$ and calculate the total sum $S_{12}$.

5. Phase 3: Simulation and decision. In a spreadsheet (Excel, Calc, or Google Sheets), create a month-by-month comparative table and a chart representing both trends. Write a brief report justifying which option you find more viable and whether the accumulated money offsets the price increase according to the recorded CPI.

Assessment criteria:

• Rigorous algebraic deduction and calculation of general terms and sums ($S_{12}$).

• Proper design of the spreadsheet, automatic formulas, and comparative chart.

• Critical and realistic reasoning for the final decision linked to the actual CPI.