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) $3, -3, 3, -3, \dots$

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.

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: $a_n = a_1 \times r^{n-1}$.

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.

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 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.

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 (downwards and upwards) 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.

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?

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?