Sequences of real numbers
2, 4, 6, 8, … what comes next? Easy: 10. And in 1, 1, 2, 3, 5, 8, …? A little harder (it is 13). Looking for the pattern in a list of numbers is like cracking a code, and it is one of the ways in which maths describes things that grow: savings, followers, figures that repeat…
What a sequence is
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)…
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:
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a) $2, 4, 6, 8, \dots$
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b) $1, 4, 9, 16, \dots$
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c) $10, 7, 4, 1, \dots$
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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: $$ \frac{1}{2},…
Not all sequences are made up of integers (like 1, 2, 3...). They can also contain fractions or decimals. Look at the following sequence of real numbers:
$$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$-
Write down the next three terms.
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Which do you think is the nth term $a_n$? Choose the correct option:
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Option A: $a_n = \frac{1}{n}$
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Option B: $a_n = \frac{1}{n+1}$
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Option C: $a_n = n + 1$
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The general term
The general term $a_n$ is a formula that gives any term from its position, without having to work out all the previous ones. It is like having the "source code" of the sequence.
If $a_n = n^2 + 1$: $\ a_1 = 2,\ a_2 = 5,\ a_3 = 10,\ \ldots$ and the 100th term is simply $a_{100} = 10\,001$.
Write a formula in terms of $n$ yourself and see what its terms look like:
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$).…
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:
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a) $a_n = 3n + 1$
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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$…
In your notebook, match each of the following sequences with its correct nth term. Note that $n$ represents the position (1, 2, 3...).
Sequences:
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A: $2, 4, 6, 8, \dots$
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B: $1, 3, 5, 7, \dots$
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C: $5, 10, 15, 20, \dots$
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D: $1, 4, 9, 16, \dots$
nth Terms:
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1: $a_n = 2n - 1$
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2: $a_n = n^2$
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3: $a_n = 2n$
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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…
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?
Recurrence relations
Sometimes the rule does not say how to calculate $a_n$ directly, but rather how to obtain each term from the previous ones. This is a recurrence relation. The most famous one is Fibonacci's: each term is the sum of the two before it.
$$a_1 = 1,\quad a_2 = 1,\quad a_n = a_{n-1} + a_{n-2} \quad\Rightarrow\quad 1, 1, 2, 3, 5, 8, 13, 21, \ldots$$Sometimes, to find the value of a term, we need to know the previous one. This is called a recurrence relation. Given a sequence where the first term is $a_1 = 5$ and…
Sometimes, to find the value of a term, we need to know the previous one. This is called a recurrence relation.
Given a sequence where the first term is $a_1 = 5$ and the rule to obtain the next terms is:
$$a_n = a_{n-1} + 3$$(This means: "the current term is equal to the previous term plus 3").
Calculate terms $a_2, a_3, a_4$ and $a_5$.
A sequence is defined by a recurrence relation when its terms are obtained by working with the previous ones. Consider the sequence defined as follows: The first term is…
A sequence is defined by a recurrence relation when its terms are obtained by working with the previous ones. Consider the sequence defined as follows:
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The first term is $a_1 = 2$.
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The second term is $a_2 = 3$.
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Any subsequent term is calculated by multiplying the two terms immediately before it: $a_n = a_{n-1} \cdot a_{n-2}$.
Calculate terms $a_3, a_4, a_5$ and $a_6$. Do you think this sequence grows quickly or slowly?
Consider the sequence defined by the following recurrence relation: $$a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2} \quad \text{for } n > 2$$ This is the famous…
Consider the sequence defined by the following recurrence relation:
$$a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2} \quad \text{for } n > 2$$This is the famous Fibonacci sequence.
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Calculate the first 10 terms of the sequence.
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Let us study the behaviour of the ratio (division) between two consecutive terms. Calculate the following values to 4 decimal places:
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$\frac{a_2}{a_1}$
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$\frac{a_3}{a_2}$
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$\frac{a_6}{a_5}$
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$\frac{a_{10}}{a_9}$
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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.
Patterns with figures
Many sequences come from growing figures. To find their formula, count what is added at each step.
An algorithm you can check and modify
An algorithm is a precise sequence of instructions. For the matchsticks we can break the problem down into three tasks: start with the first figure, add the matchsticks of each new figure and report the total. If $n$ is a positive integer:
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Set the figure counter to 1 and the total number of matchsticks to 4.
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While the counter is less than $n$, add 3 to the total and then add 1 to the counter.
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Return the total.
The table lets you follow it without programming:
| Counter | Total matchsticks | Continue if $n = 4$? |
|---|---|---|
| 1 | 4 | Yes |
| 2 | 7 | Yes |
| 3 | 10 | Yes |
| 4 | 13 | No |
To automate it in a spreadsheet, type 1 in A2, 4 in B2, =A2+1 in A3 and =B2+3 in B3. Copy the formulas down to the figure you want. In C2 type =3*A2+1 and copy it down: B and C must match in every row. Compare the repetitive method with the general term for figure 50. Both give 151; the formula avoids going through the 49 previous figures.
Your turn: invent a weekly savings sequence and ask a question that a spreadsheet could answer, such as "in which week will I exceed a certain amount?". Decide what data it needs, write the instructions and test a small case before automating it.
Increasing and decreasing sequences
A sequence is increasing if each term is greater than the previous one, decreasing if it is smaller and constant if it does not change. Some sequences are none of these, such as $(-1)^n$, which keeps jumping back and forth.
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…
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.
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Write down the first 4 terms of this sequence.
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How many dots will Figure 5 have?
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Without drawing, can you say how many dots are added to go from Figure 4 to Figure 5?
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…
Look at the following geometric pattern made with matchsticks.
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Draw Figure 4 in your notebook following the same pattern.
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Write down the numerical sequence representing the total number of matchsticks needed to build each figure ($a_1, a_2, a_3, a_4$).
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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)…
In a supermarket, an employee is stacking tin cans into a triangular structure for a display.
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In the top row (row 1) he places 1 can.
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In the row directly below (row 2) he places 3 cans.
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In row 3 he places 5 cans.
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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.