Exercises: Patterns and regularities

Observe the following number sequences and write the next two terms of each, explaining the rule they follow:

• $5,\ 9,\ 13,\ 17,\ \ldots$

• $2,\ 6,\ 18,\ 54,\ \ldots$

• $1,\ 4,\ 9,\ 16,\ \ldots$

A craft project involves building a chain of triangles with sticks, as shown in the figure.

• How many sticks are needed to build Figure 4?

• What about Figure 5?

The sequence $a_n$ has the general term $a_n = 4n - 1$. Calculate the terms $a_1$, $a_5$ and $a_{20}$.

A sequence starts at $7$ and each term is obtained by adding $6$ to the previous one. Write its first five terms and calculate the term at position $10$.

Observe the sequence of figures formed by dots in the image.

• How many dots will Figure 4 have? What about Figure 5?

• What is the relationship between the figure number and the number of dots?

The Fibonacci sequence starts like this: $1,\ 1,\ 2,\ 3,\ 5,\ 8,\ \ldots$; each term is obtained by adding the two previous ones. Write the next four terms.

Martina starts the school year with $10\ \text{€}$ in her piggy bank and, from then on, saves $3\ \text{€}$ each week.

• Write the sequence of accumulated money at the end of each of the first five weeks.

• How much money will she have at the end of week $12$?

Look at the sequence $1,\ -2,\ 3,\ -4,\ 5,\ -6,\ \ldots$

• Write the next three terms.

• Which term is in position $25$? Is it positive or negative?

Lucía observes a pattern of towers built with square blocks: tower 1 has 5 blocks, tower 2 has 8 blocks, and tower 3 has 11 blocks. Realising that 3 blocks are added at each step, she states: "Since 3 blocks are added at each stage and the first one has 5, the general term that gives the total number of blocks in tower $n$ is $a_n = 3 \cdot n + 5$".

1. a) Calculate the values given by Lucía's formula for $n = 1$, $n = 2$, and $n = 3$, and check them against the actual blocks in each tower.
2. b) Identify the reasoning error Lucía makes when relating the fixed part of the pattern to the part that increases at each step.
3. c) Deduce the correct expression for the general term $a_n$ from the structural breakdown of the figure (identifying how many fixed blocks there are and how many are added depending on the step number) and verify that it matches the three known cases.

A key principle when studying regularities is that a few initial terms do not determine a unique sequence.

1. a) Write the first four terms of the sequence whose general term is $a_n = 2^n$.
2. b) Design the general term or the formation rule of another, different sequence, $b_n$, that satisfies exactly the following two conditions: it shares the first three values with $a_n$ ($b_1 = 2$, $b_2 = 4$, and $b_3 = 8$), but its fourth term is strictly different ($b_4 \neq 16$).
3. c) Using this example as support, explain why checking that a formula works for $n = 1$, $n = 2$, and $n = 3$ does not serve as a mathematical proof that it is valid for all $n \in \mathbb{N}$.

A construction company stacks blocks to form staircases, as shown in the diagram.

• Count the blocks in figures 1, 2 and 3, and calculate how many blocks figure 4 will have.

• How many blocks will figure 10 have?

• Write a general expression for the number of blocks in figure $n$. (Hint: join each staircase with its reflection to form a rectangle.)

Consider the sequence $7,\ 12,\ 17,\ 22,\ \ldots$

• Find its general term $a_n$.

• Check whether $157$ is a term of the sequence and, if so, state its position.

• Does $200$ belong to the sequence? Justify your answer.

The figure shows how a square of $4 \times 4$ dots can be constructed by adding "L"-shaped layers.

1. a) Write the addition represented by the figure based on the number of dots in each layer, and verify the result.
2. b) Formulate a conjecture: what is the sum of the first $n$ odd numbers?
3. c) Verify your conjecture for $n = 6$.

A school amphitheatre has $12$ seats in the first row, and each subsequent row has $3$ more seats than the previous one.

• Write the number of seats in the first four rows.

• Find the general term $a_n$ that gives the number of seats in row $n$.

• If the amphitheatre has $15$ rows, how many seats are in the last row?

A sequence is defined recursively as: $a_1 = 2$ and $a_n = 2\cdot a_{n-1} + 1$.

• Calculate the first five terms.

• Compare each term with powers of $2$ ($2,\ 4,\ 8,\ 16,\ \ldots$) and describe the pattern you observe.

• Based on the pattern, what will be the value of $a_{10}$?

Look at the sequence $3,\ 6,\ 11,\ 18,\ 27,\ \ldots$

• Calculate the differences between consecutive terms. What pattern do these differences follow?

• Compare the terms with perfect squares and find the general term $a_n$.

• Calculate $a_{15}$.

In a local padel tournament, each pair plays once against every other pair.

• Complete a table showing the number of matches in the tournament if $2,\ 3,\ 4$ and $5$ pairs take part.

• Find a general expression for the number of matches with $n$ pairs. (Hint: each pair plays against $n - 1$ opponents, but counting this way counts every match twice.)

• How many matches are played if $12$ pairs take part?

A student claims that the general term of the sequence $2,\ 4,\ 8,\ 16,\ \ldots$ is $a_n = 2n$. State whether the claim is true or false, justify your answer, and write the correct general term.

On a seaside promenade, continuous benches are installed by assembling regular hexagonal concrete modules in a straight line. If a single isolated hexagonal module is placed, its 6 outer sides are free to sit on. If two modules are joined in a row, they share a common interior side (which becomes unusable), leaving 10 free sides in total. If three modules are joined in a row, there are 14 free sides left to sit on.

1. a) Without using formulas yet, make an intuitive and justified estimate: if 25 hexagonal modules are joined in a row, will the total number of free sides be closer to 80, 100, or 150? Explain what you base your reasoning on to rule out the other values.
2. b) Model the situation by deducing the general term $L_n$ that determines the number of free sides as a function of the number $n$ of joined modules, algebraically separating the sides contributed by the ends from those contributed by each intermediate module.
3. c) Calculate the exact value of $L_{25}$, compare it with your previous estimate, and assess whether the numerical result confirms that your initial reasoning was correct.

A city's municipal electric bicycle sharing service is expanding its facilities along an avenue. The first station installed has 14 charging docks. To respond to the projected increase in users towards the outskirts, the mobility plan dictates that each new station must have 4 more docks than the immediately preceding station (that is, station 2 has 18 docks, station 3 has 22, etc.).

1. a) Define the number of docks $a_n$ at station $n$ using a recurrence relation, explicitly indicating the initial term $a_1$ and the relation that allows obtaining $a_n$ from $a_{n-1}$.
2. b) Deduce the explicit general term $a_n$ that allows calculating directly how many docks any station $n$ will have without needing to calculate all intermediate stations.
3. c) The city council receives a fixed budget allocation with exactly 70 docks to equip a single station on the avenue. Set up the corresponding equation with the general term and solve it to check whether there will be any station that uses exactly those 70 docks or if material will be left over; interpret the result in the context of the problem.

The dining hall at a camp joins square tables in a row to seat participants, as shown in the diagram.

• How many people can be seated if $4$ tables are joined?

• Find the general term giving the number of diners for $n$ tables.

• A total of $42$ people attend the closing dinner. What is the minimum number of tables that must be joined to seat everyone?

Two classmates are discussing the general term of the sequence that starts $2,\ 4,\ 8,\ \ldots$

• Aitana says it is $a_n = 2^n$.

• Bruno says it could be $b_n = n^2 - n + 2$.

• Check that both expressions generate the first three terms.

• Calculate the fourth term using each of the two expressions. Do they match?

• Draw conclusions: do the first three terms of a sequence uniquely determine its general term?

Look at the sequence of fractions $\frac{1}{2},\ \frac{2}{3},\ \frac{3}{4},\ \frac{4}{5},\ \ldots$

• Find its general term $a_n$.

• Calculate $a_{10}$ and $a_{100}$ correct to three decimal places.

• The terms keep increasing, but could any term ever be greater than $1$? Justify your answer by looking at the expression for the general term.

Zeno of Elea (5th century BC) pondered infinite sums. Imagine a square of side length $1$ with half of it shaded; then half of the remaining part; then half of what remains, and so on.

• Write the fraction of the square that has been shaded after each of the first four steps.

• Find the general term $s_n$ for the shaded fraction after $n$ steps. (Hint: notice that after each step $\frac{1}{2^n}$ of the square remains unshaded.)

• If the process continued indefinitely, what value would the shaded area approach?

In his book Liber Abaci (1202), Leonardo of Pisa, known as Fibonacci, posed this problem: a pair of newborn rabbits takes one month to become adults; from then on, every month they produce a new pair of offspring, which in turn follow the same rule. No rabbits die.

• Starting from a single newborn pair, complete a table showing the number of pairs at the end of each of the first six months.

• Write down the recurrence relation followed by the sequence.

• How many pairs will there be after $10$ months?

Pascal's triangle (also known as Tartaglia's triangle) is built by placing ones along the edges and filling each interior box with the sum of the two numbers above it:

• Add up the numbers in each of the five rows and write down the sequence of these sums.

• Find the general term of this sequence.

• What is the sum of the numbers in row $12$?

Triangular numbers are $1,\ 3,\ 6,\ 10,\ 15,\ \ldots$ (dots forming a triangle) and square numbers are $1,\ 4,\ 9,\ 16,\ 25,\ \ldots$

• Check using the first four cases that every square number (from $4$ onwards) is the sum of two consecutive triangular numbers.

• Write this property as an algebraic identity using the general terms $T_n = \frac{n\cdot(n + 1)}{2}$ and $C_n = n^2$, and prove it.

• Draw a dot pattern to illustrate why this property holds for $C_4 = 16$.

The Collatz conjecture, proposed in 1937 and still unproven, starts with any natural number and applies this rule: if the number is even, divide it by $2$; if it is odd, multiply it by $3$ and add $1$. The conjecture states that you will always eventually reach $1$.

• Apply the rule starting from $7$ and write out the full sequence until reaching $1$.

• Do the same starting from $11$.

• Test the conjecture for three numbers of your choice, one of which is greater than $50$. Do you think testing many cases proves that the conjecture is true? Comment on your answer.

For the decorative lighting of an auditorium, a triangular panel of light bulbs is designed: the first row has 1 bulb, the second row has 2 bulbs, the third has 3 bulbs, and so on up to row $n$, which contains $n$ bulbs. The goal is to calculate the total number of bulbs, $T_n$, that make up a triangular structure of $n$ rows.

Two technicians propose different methods to solve the task:

• Marcos suggests a recurrence relation: taking $T_1 = 1$ and $T_n = T_{n-1} + n$, calculating row by row cumulatively.

• Elena proposes a geometric method: mentally doubling the figure with another identical, inverted triangle to form between the two a rectangle of dimensions $n \times (n+1)$ bulbs, calculating $T_n$ as half the total of that rectangle.

1. a) Algebraically develop Elena's proposal to obtain the explicit expression for $T_n$ in terms of $n$.
2. b) Apply both methods to calculate how many bulbs are needed in a panel of $n = 60$ rows and check whether both lead to the same result.
3. c) Critically compare the two alternatives: if the city council has a batch of 1830 bulbs and asks you to find out how many exact rows the panel that can be built will have, which of the two methods is more efficient and direct to answer? Also justify in which step-by-step design or programming scenario Marcos's approach would be more practical than Elena's.

A telecommunications company deploys a network of repeaters in a technology park. To ensure link reliability in the event of potential failures, each repeater must be connected to every other repeater via an independent, direct optical fibre channel.

• With 2 repeaters, 1 direct channel is needed.

• With 3 repeaters, 3 direct channels are needed.

• With 4 repeaters, 6 direct channels are needed.

• With 5 repeaters, 10 direct channels are needed.

An installer observes these four data points and states: "Since the pattern works for 2, 3, 4, and 5 repeaters, I conjecture that the formula for $n$ repeaters is $C_n = \dfrac{n^2 - n}{2}$, but I cannot be sure whether it will fail when we have 30 repeaters".

1. a) Algebraically prove that the conjecture is true for any natural number $n \ge 2$: explain how many channels originate from each of the $n$ repeaters and clearly justify why it is essential to divide the resulting product by 2.
2. b) Based on your proof, explain the difference between a numerical check of particular cases and a rigorous algebraic proof of a pattern for all $n$.
3. c) If the allocated technical budget only allows financing at most 136 optical fibre channels, set up a quadratic equation and calculate the maximum number of repeaters that can be completely interconnected with one another without exceeding the budget limit.