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 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 $10$.

Martina starts the school year with $\$200$ in her piggy bank and, from then on, saves $\$60$ 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?

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?

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 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?

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

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.

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 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.

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?

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?

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?

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 $12$?

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?

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 $T_n = \frac{n\times(n + 1)}{2}$ 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.