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

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: $a_n = n + 1$

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: $a_n = 5n$

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?

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.

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.

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?

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.

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.