Exercises: Sequences of real numbers

Look at the following ordered lists of numbers. Your task is to identify the logical rule behind each one and write down the next three terms in the sequence:

• a) $2, 4, 6, 8, \dots$

• b) $1, 4, 9, 16, \dots$

• c) $10, 7, 4, 1, \dots$

• d) $3, 6, 12, 24, \dots$

In mathematics, we use formulas to define the value at any position in a sequence without having to write out the entire list. This is called the nth term ($a_n$).

Calculate the first four terms ($a_1, a_2, a_3, a_4$) and the 10th term ($a_{10}$) for the following sequences defined by their nth term:

• a) $a_n = 3n + 1$

• b) $b_n = n^2 - 2$

Sequences often arise from geometric patterns. Look at the following image showing how a triangle formed by dots grows.

Count the number of dots in each figure to form a numerical sequence.

1. Write down the first 4 terms of this sequence.

2. How many dots will Figure 5 have?

3. Without drawing, can you say how many dots are added to go from Figure 4 to Figure 5?

Imagine you are training for a long-distance race. Your coach notes down the kilometres you run each week to track your progress. The list for the last 6 weeks is as follows:

Considering this list as a numerical sequence ($a_n$):

1. What is the value of the term $a_1$? What does it represent in the context of training?

2. What is the value of the term $a_4$?

3. If you continue at the same rate of increase, how many kilometres will you run in week 7 ($a_7$)?

In your notebook, match each of the following sequences with its correct nth term. Note that $n$ represents the position (1, 2, 3...).

Sequences:

• A: $2, 4, 6, 8, \dots$

• B: $1, 3, 5, 7, \dots$

• C: $5, 10, 15, 20, \dots$

• D: $1, 4, 9, 16, \dots$

nth Terms:

• 1: $a_n = 2n - 1$

• 2: $a_n = n^2$

• 3: $a_n = 2n$

• 4: $a_n = 5n$

Sometimes, to find the value of a term, we need to know the previous one. This is called a recurrence relation.

Given a sequence where the first term is $a_1 = 5$ and the rule to obtain the next terms is:

(This means: "the current term is equal to the previous term plus 3").

Calculate terms $a_2, a_3, a_4$ and $a_5$.

A video game "streamer" decides to start saving up to buy new equipment.

In the first month they save €50, in the second month they save €100, in the third month they save €200 and in the fourth month they save €400.

1. Write the sequence formed by the money saved each month.

2. Do you notice any pattern? What operation do you do to go from one month to the next?

3. Calculate how much they will save in month 5 and in month 6 if they keep up this trend.

Not all sequences are made up of integers (like 1, 2, 3...). They can also contain fractions or decimals. Look at the following sequence of real numbers:

1. Write down the next three terms.

2. Which do you think is the nth term $a_n$? Choose the correct option:

   • Option A: $a_n = \frac{1}{n}$

   • Option B: $a_n = \frac{1}{n+1}$

   • Option C: $a_n = n + 1$

Aoife wants to find out if the number 39 is part of the sequence whose general term is $a_n = 5n - 6$. On her exam, she writes the following:

“To check, I calculate $a_{39} = 5 \times 39 - 6 = 195 - 6 = 189$. Since the operation gives an exact result, I conclude that the number 39 does belong to the sequence and occupies position 189”.

1) Identify and explain in detail the conceptual error Aoife made by confusing the position index $n$ with the value of the term $a_n$.

2) Set up and solve the appropriate equation to determine whether the number 39 really belongs to the sequence, indicating which position it occupies if the answer is affirmative.

3) Reason whether the number 50 belongs to this same sequence or not, taking into account what type of numbers the position indices $n$ must necessarily be.

Design the general term $a_n$ of a sequence of real numbers that simultaneously satisfies these three conditions:

1. Its first term is a specific fraction: $a_1 = \frac{1}{2}$.

2. All terms are strictly positive and each term is less than the previous one (that is, $a_1 > a_2 > a_3 > \dots > 0$).

3. As the position $n$ becomes very large, the terms get closer and closer to 0, but without ever becoming zero or taking negative values.

Write the algebraic expression for your general term $a_n$, verify that it satisfies the conditions by explicitly calculating the terms $a_1, a_2, a_3$, and $a_4$, and briefly justify why no term of your formula can ever be equal to 0.

Given the sequence whose nth term is given by the algebraic expression $a_n = \frac{n^2 - 1}{n + 1}$, calculate the first four terms and the 15th term. Do you notice anything interesting when simplifying the initial algebraic expression?

Below are several numerical sequences that follow a logical pattern. Your task is to identify the rule of the sequence and write down the next three terms for each one:

• a) $4, \ 7, \ 10, \ 13, \ ...$

• b) $3, \ -6, \ 12, \ -24, \ ...$

• c) $1, \ 4, \ 9, \ 16, \ 25, \ ...$

• d) $\frac{1}{2}, \ \frac{2}{3}, \ \frac{3}{4}, \ \frac{4}{5}, \ ...$

Look at the following geometric pattern made with matchsticks.

1. Draw Figure 4 in your notebook following the same pattern.

2. Write down the numerical sequence representing the total number of matchsticks needed to build each figure ($a_1, a_2, a_3, a_4$).

3. Describe in your own words how the number of matchsticks increases from one figure to the next.

You can do it on paper and send a photo with your answer.

A sequence is defined by a recurrence relation when its terms are obtained by working with the previous ones. Consider the sequence defined as follows:

• The first term is $a_1 = 2$.

• The second term is $a_2 = 3$.

• Any subsequent term is calculated by multiplying the two terms immediately before it: $a_n = a_{n-1} \times a_{n-2}$.

Calculate terms $a_3, a_4, a_5$ and $a_6$. Do you think this sequence grows quickly or slowly?

In a supermarket, an employee is stacking tin cans into a triangular structure for a display.

• In the top row (row 1) he places 1 can.

• In the row directly below (row 2) he places 3 cans.

• In row 3 he places 5 cans.

• In row 4 he places 7 cans.

Write down the sequence that represents the number of cans per row and find a general formula ($a_n$) that allows us to find out how many cans there will be in row $n$ without having to count them one by one.

Find the nth term ($a_n$) of the following sequences. Note that some may be arithmetic, geometric, or based on square numbers:

• a) $5, \ 10, \ 15, \ 20, \ ...$

• b) $3, \ 6, \ 12, \ 24, \ ...$

• c) $\frac{1}{2}, \ \frac{1}{4}, \ \frac{1}{8}, \ \frac{1}{16}, \ ...$

• d) $1, \ 8, \ 27, \ 64, \ ...$

Aoife is training for a marathon. On the first day she runs for 15 minutes. Her coach has set out a plan where each day she must run 4 minutes longer than the previous day.

1. Write down the sequence representing the number of minutes Aoife runs during the first 5 days.

2. If she maintains this training plan, how many minutes will she have to run on day 20? Use the nth term formula to calculate it.

We have a sequence whose nth term is $b_n = n^2 - 3n + 2$.

1. Calculate the first 5 terms.

2. Is there any term in the sequence equal to 0? If so, which position or positions does it occupy? Justify your answer by solving the corresponding equation.

In Galway, the municipal electric bike rental service applies the following rate for non-subscribed users: unlocking the bike incurs a base fare of $0.70$ €, which covers the first 15 minutes of the ride. From then on, for each new 15-minute section (or fraction thereof) that begins, $0.40$ € is added to the total cost.

We define the sequence $C_n$ as the total amount in euros to be paid for a trip that lasts $n$ sections of 15 minutes (where $n = 1$ corresponds to a trip of up to 15 minutes, $n = 2$ to a trip between 16 and 30 minutes, etc.).

1) Write the first four terms of the sequence: $C_1, C_2, C_3$, and $C_4$.

2) Deduce the algebraic expression of the general term $C_n$ in terms of the number of sections $n$.

3) A user checks the bank statement in the app and sees that they have been charged exactly $3.50$ € for a single trip. Set up the corresponding equation with the general term, find the value of $n$, and interpret the result by indicating between which minutes the duration of their trip fell.

A sequence of real numbers is defined by the following recurrence relation: its first term is $a_1 = 3$ and each subsequent term is obtained by multiplying the previous one by 2 and subtracting 1, that is, $a_n = 2 \times a_{n-1} - 1$ for all $n \ge 2$.

1) Estimate: Without performing long calculations on paper, estimate intuitively whether the tenth term ($a_{10}$) will exceed the value of $1\,000$. Briefly justify the reasoning behind your prediction.

2) Calculate: Obtain the first five terms of the sequence ($a_1, a_2, a_3, a_4, a_5$) in an orderly manner by applying the recurrence relation step by step.

3) Deduce and check: Compare the values obtained with the powers of two ($2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, $2^5 = 32, \dots$). Deduce a closed-form formula for the general term $a_n$ in terms of $n$, use it to calculate the exact value of $a_{10}$, and evaluate whether your initial estimate was correct.

Look at the following geometric pattern formed by small squares.

• Figure 1 is a single central square.

• Figure 2 is formed by adding a square to each free side of the previous figure (top, bottom, left, right).

• Figure 3 follows the same pattern, adding squares only to the outer free sides of the previous figure.

Tasks:

1. Draw Figure 4 in your notebook.

2. Write down the sequence formed by the total number of squares in the first 5 figures.

3. Explain your reasoning and find an algebraic expression for the nth term $a_n$ that allows you to calculate the number of squares in the figure at position $n$.

4. Calculate how many squares Figure 20 would have without drawing it.

You can do it on paper and send a photo with your answer.

Study the following sequences of real numbers. You must identify the logical pattern and find the nth term $a_n$ in terms of $n$ for each of them.

• Sequence A: $0, \frac{3}{2}, \frac{8}{3}, \frac{15}{4}, \frac{24}{5}, \dots$

• Sequence B: $-2, 4, -8, 16, -32, \dots$

• Sequence C: $2, 5, 10, 17, 26, \dots$

In computer science and algorithm design, it is common to come across the Collatz sequence, also known as the $3n+1$ conjecture. Although it seems simple, it is one of the most famous open problems in mathematics.

The formation rule is recurrent. Given an initial natural number $a_1$, the following terms are obtained as follows:

• If the previous term $a_{n-1}$ is even, the next one is half of it: $a_n = \frac{a_{n-1}}{2}$.

• If the previous term $a_{n-1}$ is odd, the next one is obtained by multiplying by 3 and adding 1: $a_n = 3 \times a_{n-1} + 1$.

• Write the complete sequence starting with $a_1 = 13$ until you reach the number 1.

• How many terms does the generated sequence have (length of the orbit) until it reaches 1 for the first time?

• Check what happens if you keep applying the rule once you have reached 1. Describe the cycle that forms.

We have a regular polygon with $n$ sides. We want to study the sequence formed by the total number of diagonals of the polygon as its number of sides increases.

• $a_3$ (triangle, 3 sides): 0 diagonals.

• $a_4$ (square, 4 sides): 2 diagonals.

• $a_5$ (pentagon, 5 sides): 5 diagonals.

• Draw a regular hexagon ($n=6$) and draw all its diagonals. What is the value of $a_6$?

• Notice that to go from $a_3$ to $a_4$ we add 2; from $a_4$ to $a_5$ we add 3; from $a_5$ to $a_6$ we add 4. Following this growth pattern, calculate how many diagonals a decagon ($n=10$) will have.

• The nth term of this sequence is given by the quadratic formula $a_n = \frac{n(n-3)}{2}$. Use this formula to check your answer from the previous part.

You can do it on paper and send a photo with your answer.

Consider the sequence defined by the following recurrence relation:

This is the famous Fibonacci sequence.

1. Calculate the first 10 terms of the sequence.

2. Let us study the behaviour of the ratio (division) between two consecutive terms. Calculate the following values to 4 decimal places:

   • $\frac{a_2}{a_1}$

   • $\frac{a_3}{a_2}$

   • $\frac{a_6}{a_5}$

   • $\frac{a_{10}}{a_9}$

3. Do you notice that the results get closer to a specific number? Research which famous irrational number is approached when dividing consecutive terms of this sequence when $n$ is very large.

Imagine a 2D platform video game. The character is at the bottom of a staircase with $n$ steps.

The character has two ways to climb:

• Climb 1 step with a regular jump.

• Climb 2 steps with a super-jump.

We want to know how many different ways ($F_n$) the character can reach the top of the staircase.

• If there is 1 step ($n=1$): There is only 1 way (1-step jump). $F_1 = 1$.

• If there are 2 steps ($n=2$): They can do (1+1) or (2). There are 2 ways. $F_2 = 2$.

• If there are 3 steps ($n=3$): They can do (1+1+1), (1+2) or (2+1). There are 3 ways. $F_3 = 3$.

• Calculate how many different ways they can climb a staircase with 4 steps ($F_4$) and 5 steps ($F_5$). Describe them (for example: 1+2+1+1).

• Find the relationship between $F_n$, $F_{n-1}$ and $F_{n-2}$.

• If the staircase had 10 steps, in how many different ways could they climb it?

The Sierpinski triangle is defined by an iterative (repeated) process of area removal.

We start with a black equilateral triangle of area $A = 64 \text{ cm}^2$ (Iteration 0).

• Iteration 1: We connect the midpoints of the sides and cut out the inverted central triangle (this white triangle is removed). We are left with 3 smaller black triangles.

• Iteration 2: We repeat the process with each of the 3 remaining black triangles.

Tasks:

1. Write down the sequence $T_n$ representing the number of black triangles at each step (step 0, step 1, step 2, step 3). What type of sequence is it?

2. Write down the sequence $S_n$ representing the total black area remaining in the figure at each step.

3. Calculate what black area will remain in the figure after performing iteration 5.

Observe the following sequence of repeating decimals:

Although it seems that you are simply "adding ones", let's analyze its fractional value.

1. Write the first three terms of the sequence as irreducible fractions.

2. Find the general term $a_n$ expressed as a fraction.

3. Calculate the sum of the first two terms $a_1 + a_2$ and express the result as a fraction.

4. Investigate and reason: Does the term $a_n$ change when $n$ is a very large number (for example, with 100 ones) or does it remain constant? Justify your answer.

To calculate the term occupying position number 80 ($a_{80}$) in the sequence that begins with $5, 11, 17, 23, 29, \dots$, two 3rd Year students propose different approaches:

• Method A (Hugo): Defines the sequence using a recurrence relation: $a_1 = 5$ and $a_n = a_{n-1} + 6$, and proposes using a spreadsheet or calculating each term successively by hand until reaching position 80.

• Method B (Aoife): Looks for a direct algebraic expression for the general term $a_n$ in terms of the position $n$, and calculates $a_{80}$ by directly substituting $n = 80$.

1) Apply Aoife's method: deduce with reasoning the formula for the general term $a_n$ from the regularity of the given terms and calculate the exact value of $a_{80}$.

2) Critically compare both methods: in what practical situation would it be preferable or more intuitive to use Hugo's method, and why is Aoife's method significantly more advantageous to answer the question posed?

3) Analyze the Fibonacci sequence ($1, 1, 2, 3, 5, 8, 13, \dots$). Explain why in 3rd Year it is not feasible to use Aoife's method with this sequence and it is essential to resort to the recurrence relation.

A biomedical laboratory in Granada analyzes the proliferation of a cell culture arranged in a dish with a concentric geometric structure:

• In the initial stage ($n = 1$), there is a single central cell ($c_1 = 1$).

• In stage $n = 2$, a first ring with 8 new cells forms around the central cell ($c_2 = 8$).

• In stage $n = 3$, the next outer ring requires 16 new cells ($c_3 = 16$).

• In stage $n = 4$, the next outer ring needs 24 new cells ($c_4 = 24$).

Let $c_n$ be the sequence indicating the number of new cells added exclusively to the outer ring of stage $n$ (for $n \ge 2$).

1) Write the recurrence relation that connects $c_n$ with $c_{n-1}$ for $n \ge 3$, and find the direct algebraic expression of the general term $c_n$ in terms of $n$ valid for all $n \ge 2$.

2) Inverse problem: The laboratory microscope only allows an outer ring to be observed clearly if it has at most 200 cells. Set up the corresponding inequality, calculate what is the furthest stage $n$ that can be observed completely, and determine how many cells exactly that outer ring will have.

3) Algebraic proof: A researcher conjectures that the sum of new cells from any two consecutive outer rings, $c_n + c_{n+1}$ (with $n \ge 2$), is always a multiple of 8 but can never be a multiple of 16. Prove algebraically whether the researcher's statement is true or false by analyzing the resulting expression of this sum.

Topic challenge. How far does a bounce go? Model the physics of your favorite ball by applying sequences of real numbers.

1. Data collection: Drop a ball from a height of $100\text{ cm}$ next to a wall marked with a tape measure. Record it in slow motion with your phone and note the maximum height reached on the first 4 bounces ($a_1, a_2, a_3, a_4$).

2. Modeling: In a spreadsheet or GeoGebra, record your data and calculate the quotients between consecutive terms ($a_{n}/a_{n-1}$) to find the average common ratio $r$ (use decimals with a comma, such as $0.75$). Write the general term of your progression: $a_n = a_1 \times r^{n-1}$.

3. Analysis and prediction: Predict the height of the eighth bounce and calculate the infinite sum of the heights $S = \frac{a_1}{1-r}$ and, from it, the total theoretical vertical distance traveled during the bounces, $2S$ (each bounce is one rise and one fall).

4. Final report: Create a brief document with your table, graph of the sequence, and a reasoned conclusion: why does the ball stop in practice sooner than the mathematical model predicts?

Assessment criteria:

• Rigor in obtaining the common ratio $r$, the formula for the general term, and the sum.

• Correct use of the spreadsheet or GeoGebra to tabulate and plot the sequence.

• Critical justification of the differences between the theoretical model and physical reality.