Exercises: Geometric progressions

Analyze the following number sequences. Your task is to identify which of them are geometric progressions. For those that are, indicate 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 of side $8 \text{ cm}$. The next square is formed with a side length that is half of 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 that a viral video on a social network is shared very quickly. One person uploads the video (minute 0). In minute 1, 5 new people watch it. In minute 2, each of those 5 people shares it with 5 other new people, and so on.

1. Write 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 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 7th term ($a_7$).
2. b) Calculate the value of the 10th term ($a_{10}$).

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

A grandfather decides to save for his grandson following a curious mathematical rule. The first month he puts $\$2$ in a piggy bank. The second month he puts in $\$6$. The third month he puts in $\$18.$

If he continues with this logic:

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

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

We have a geometric progression made up of fractions. We know that the first terms are:

$$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots $$

1. a) Identify the first term $a_1$ and the common ratio $r$.
2. b) Calculate the term $a_6$ without multiplying one by one, 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, every hour that passes, the number of new views is multiplied by 3 compared to the previous hour.

1. a) Write 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 you to calculate the views in hour $n$.
4. d) How many new views will the video have exactly in hour 8?

In a role-playing video game, the experience needed to level up follows a geometric progression. We know that to reach level 2 you need 150 experience points, and to reach level 5 you need 1200 points.

Calculate:

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

Look at the following fractal figure. We start with an equilateral triangle of area 64 sq in (Stage 1). In each subsequent step, in each of the triangles that remain uncolored, we color the central triangle formed by the midpoints of its sides (inverted triangle), which has an area equal to $1/4$ of the triangle where it is located.

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

1. a) Write 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 colored triangles if we repeated the process an infinite number of times.

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

1. a) How much money will he have to save in week number 8?
2. b) How much money will he have accumulated 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 feet. On each bounce, the ball loses energy and only reaches 80% of the previous height ($r = 0,8$).

We have a square with a side length of 10 inches. 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 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 that the fourth term is $a_4 = 24$.

Hint: First you will need to use the data to find the common ratio $r$ of the progression.

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 hours elapsed $n$.

2. Calculate exactly how many bacteria there will be 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 feet. Due to energy loss, every time it bounces on the ground, it rises to only 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 the ball reaches after the fifth bounce.

2. Imagine we let the ball 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 is unique, but the bounces involve going up and down the same distance.

A famous YouTuber launches a charity campaign. On the first day he donates $\$10.$ He commits that on each subsequent day he will donate an amount such that the total sum donated up to that day is triple what he donated individually the previous day.

Note: Read the question carefully, it is a mathematical riddle about the common ratio $r$.

1. Calculate how much he donates on the second and third day.

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

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

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

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

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

1. Use the Pythagorean theorem to find the side of the second square ($L_2$) and the third ($L_3$).

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

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

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

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

• Contract B: Initial salary of $\$22,000$ per year, with an 8% increase each year compared to the previous year (Geometric Progression).

• Write the general 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 sum of money earned in the 5 years with each contract ($S_5$). Which contract is more beneficial in the long term (in the accumulated total)?

We have a geometric progression of which we do not know the common ratio $r$ and 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 formula for the general term 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 constructed from an equilateral triangle.

The process is as follows:

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

• 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).

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

• What is the common ratio $r$ of this decreasing geometric progression?

• Calculate what black area will remain after applying the process 5 times (Step 5).

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

$$ 3, \quad x_1, \quad x_2, \quad x_3, \quad x_4, \quad 96 $$

1. Identify which position 3 and 96 occupy in the progression ($a_1$ and $a_n$).

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

3. Write the four numbers we have interpolated.

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