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 millimeters.

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 40 pesos into a piggy bank. In the second month he puts 120 pesos. In the third month he puts 360 pesos.

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.

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.

Miguel wants to buy an electric scooter that costs 8 000 pesos. He decides to save up by following a strict plan: in the first week he saves 40 pesos, in the second week 80 pesos, in the third 160 pesos, 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 drop is 20 m, but after that 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 $a_4 = 24$.

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 meters. 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 behavior 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 (downward and upward) that the ball will have traveled 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 pesos. 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 pesos per year, with a fixed increase of 2 000 pesos each year (Arithmetic Progression).

• Contract B: Starting salary of 22 000 pesos 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?