Exercises: Sequences of real numbers

Observe the following ordered lists of numbers. Your task is to identify the logical rule that each one follows and write the three terms that continue the series:

• a) $2, 4, 6, 8, \dots$

• b) $1, 4, 9, 16, \dots$

• c) $10, 7, 4, 1, \dots$

• d) $3, 6, 12, 24, \dots$

In mathematics, we use formulas to define the value of any position in a sequence without having to write the whole list. This is called the general 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 general term:

• a) $a_n = 3n + 1$

• b) $b_n = n^2 - 2$

Sequences often arise from geometric patterns. Observe 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 the first 4 terms of this sequence.

2. How many dots will figure 5 have?

3. Without drawing, could you say how many dots are added to go from figure 4 to figure 5?

Imagine you are training for a long-distance race. Your coach notes down the miles you run each week to keep track of 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 the training?

2. What is the value of the term $a_4$?

3. If you follow the same rate of increase, how many miles will you run in week 7 ($a_7$)?

Match in your notebook each of the following sequences with its correct general term. Bear in mind 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: $1, 4, 9, 16, \dots$

General Terms:

• 1: $a_n = 2n - 1$

• 2: $a_n = n^2$

• 3: $a_n = 2n$

• 4: $a_n = 5n$

Sometimes, to know 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 following ones is:

(This means: "the current term is equal to the previous term plus 3").

Calculate the terms $a_2, a_3, a_4$ and $a_5$.

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

The first month he saves $\$50$, the second month he saves $\$100$, the third month he saves $\$200$ and the fourth month he saves $\$400$.

1. Write the sequence formed by the money saved each month.

2. Do you notice any regularity? What operation do you do to get from one month to the next?

3. Calculate how much he will save in month 5 and month 6 if he keeps this trend.

Not all sequences are formed by integers (like 1, 2, 3...). They can also have fractions or decimals. Look at the following sequence of real numbers:

1. Write the next three terms.

2. What do you think is the general term $a_n$? Choose the correct option:

   • Option A: $a_n = \frac{1}{n}$

   • Option B: $a_n = \frac{1}{n+1}$

   • Option C: $a_n = n + 1$

Given the sequence whose general 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 curious when simplifying the initial algebraic expression?

Below are several number sequences that follow a logical pattern. Your task is to identify the rule of formation and write the next three terms for each one:

• a) $4, \ 7, \ 10, \ 13, \ ...$

• b) $3, \ -6, \ 12, \ -24, \ ...$

• c) $1, \ 4, \ 9, \ 16, \ 25, \ ...$

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

Look at the following geometric construction made of matchsticks.

1. Draw Figure 4 in your notebook following the same construction logic.

2. Write the number sequence that represents 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 when going from one figure to the next.

A sequence is defined by a recurrence relation when its terms are obtained by operating on 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 immediately preceding terms: $a_n = a_{n-1} \cdot a_{n-2}$.

Calculate the terms $a_3, a_4, a_5$ and $a_6$. Do you think this sequence grows quickly or slowly?

In a supermarket, an employee is stacking cans forming a triangular structure for a promotion.

• In the top row (row 1) he places 1 can.

• In the row immediately below (row 2) he places 3 cans.

• In row 3 he places 5 cans.

• In row 4 he places 7 cans.

Write the sequence that represents the number of cans per row and find a general formula ($a_n$) that allows us to know how many cans there will be in row $n$ without having to count them one by one.

Find the general term ($a_n$) of the following sequences. Bear in mind that some may be arithmetic, geometric or perfect squares:

• a) $5, \ 10, \ 15, \ 20, \ ...$

• b) $3, \ 6, \ 12, \ 24, \ ...$

• c) $\frac{1}{2}, \ \frac{1}{4}, \ \frac{1}{8}, \ \frac{1}{16}, \ ...$

• d) $1, \ 8, \ 27, \ 64, \ ...$

Ashley is training for a marathon. On the first day she runs for 15 minutes. Her coach has prepared a plan where each day she must run 4 minutes more than the previous day.

1. Write the sequence that represents the minutes Ashley runs during the first 5 days.

2. If she keeps to this training plan, how many minutes will she have to run on day 20? Use the general term to calculate it.

We have a sequence whose general term is $b_n = n^2 - 3n + 2$.

1. Calculate the first 5 terms.

2. Is there any term in the sequence that equals 0? If so, what position or positions does it occupy? Justify your answer by solving the corresponding equation.

Look at the following construction of geometric figures made of 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 free outer sides of the previous figure.

You are asked to:

1. Draw Figure 4 in your notebook.

2. Write the sequence formed by the total number of squares in the first 5 figures.

3. Reason and find an algebraic expression for the general term $a_n$ that allows you to calculate the number of squares in the figure in position $n$.

4. Calculate how many squares Figure 20 would have without needing to draw it.

Study the following sequences of real numbers. You must identify the logical pattern of construction and find the general 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: $2, 5, 10, 17, 26, \dots$

Hint for Sequence A: Look separately at how the numerator evolves and how the denominator evolves. Try to relate the numerator to the square of the position $n$.

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 rule of formation is recurrent. Given an initial natural number $a_1$, the following terms are obtained like this:

• 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: $a_n = 3 \cdot a_{n-1} + 1$.

• Write the complete sequence starting with $a_1 = 13$ until you reach the number 1.

• How many terms does the generated sequence have (orbit length) until it reaches 1 for the first time?

• Check what happens if you continue applying the rule once you have reached 1. Describe the cycle that is formed.

We have a regular polygon with $n$ sides. We want to study the sequence formed by the total number of diagonals this polygon has 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 get 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 general term of this sequence is given by the quadratic formula $a_n = \frac{n(n-3)}{2}$. Use this formula to check your result from the previous section.

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. We are going to study the behavior of the quotient (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 are getting closer to a specific number? Investigate which famous irrational number is approximated by dividing consecutive terms of this sequence when $n$ is very large.

Imagine a 2D platform video game. The character is at the bottom of a staircase with $n$ steps.

The character has two ways to go up:

• Go up 1 step with a jump.

• Go up 2 steps with a super-jump.

We want to know in how many different ways ($F_n$) he can reach the top of the stairs.

• If there is 1 step ($n=1$): There is only 1 way (jump of 1). $F_1 = 1$.

• If there are 2 steps ($n=2$): He can do (1+1) or (2). There are 2 ways. $F_2 = 2$.

• If there are 3 steps ($n=3$): He can do (1+1+1), (1+2) or (2+1). There are 3 ways. $F_3 = 3$.

• Calculate in how many different ways he can go up a staircase of 4 steps ($F_4$) and 5 steps ($F_5$). Describe them (for example: 1+2+1+1).

• Find the relationship that exists between $F_n$, $F_{n-1}$ and $F_{n-2}$.

• If the staircase had 10 steps, in how many different ways could he go up?

The Sierpinski triangle is defined by an iterative process (which repeats) of removing areas.

We start with a black equilateral triangle of area $A = 64 \text{ cm}^2$ (Iteration 0).

• Iteration 1: We join the midpoints of the sides and cut out the inverted central triangle (that 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.

You are asked to:

1. Write the sequence $T_n$ that represents the number of black triangles at each step (step 0, step 1, step 2, step 3). What type of sequence is it?

2. Write the sequence $S_n$ that represents the total black area remaining in the figure at each step.

3. Calculate what black area will remain in the figure after performing iteration number 5.

Look at the following sequence of repeating decimals:

Although it seems that we simply "add ones", let's analyze their fractional value.

1. Write the first three terms of the sequence as an irreducible fraction.

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: What integer does the term $a_n$ closely approximate when $n$ is a very large number (for example, with 100 ones)? Hint: think about the generating fraction of $0,\widehat{9}$.