Patterns and regularities

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

• $5,\ 9,\ 13,\ 17,\ \ldots$

• $2,\ 6,\ 18,\ 54,\ \ldots$

• $1,\ 4,\ 9,\ 16,\ \ldots$

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?

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?

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?

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?

Aoife 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 $a_n = 3 \times n + 5$".

1. a) Calculate the values given by Aoife'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 Aoife 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 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.)

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

Aoife starts the school year with $€10$ in her piggy bank and, from then on, saves $€3$ 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 $12$?

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.

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 $a_{15}$.

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 $10$.

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.

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

• 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 $a_{10}$?

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?

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 $n = 6$.

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?

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.