Number sequences and arithmetic progressions

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) $3, 6, 12, 24, \dots$

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: $a_n = n + 1$

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

Imagine you are training for a long-distance race. Your coach notes down the kilometers 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 kilometers will you run in week 7 ($a_7$)?

María 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 María 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.

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) $b_n = n^2 - 2$

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: $a_n = 5n$

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) $1, \ 8, \ 27, \ 64, \ ...$

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?

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.

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 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?

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 behavior 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.

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.

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?

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?

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.

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.

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.

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

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

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.

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: $2, 5, 10, 17, 26, \dots$

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.

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.