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 millimetres.

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)?

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.

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 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.

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.

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.

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?

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.