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Investigating and extending number patterns

4.2 Investigating and extending number patterns

Kinds of numeric patterns

A list of numbers that form a pattern is called a sequence. Each number in a sequence is called a term of the sequence. The first number is the first term of the sequence.

sequence
a list of numbers that follow each other in a particular order to create a pattern

We can look for a pattern or relationship between consecutive terms in order to extend the given pattern. For example, the numbers \(2; 5; 8; 11; 14; 17; 20; 23\) form a sequence, because there is a pattern in the way the numbers are ordered: we add \(3\) to the previous term. The first term of this sequence is \(2\).

consecutive
numbers or terms that follow one another in order

There are many different number patterns. Here are some of the common number patterns you will work with in Grade 8.

Adding or subtracting the same number

When the differences between consecutive terms of a sequence are the same, we say the difference is constant, or that the sequence has a common difference.

For example, if we start with \(14\) and subtract \(\mathbf{3}\) each time, we get:

\[\begin{align} 14 - 3 &= 11 \\ 11 - 3 &= 8 \\ 8 - 3 &= 5 \\ 5 - 3 &= 2 \\ \end{align}\]

The number pattern that is formed is {\(14; 11; 8; 5; 2\)}.

If we start with \(14\) and add \(\mathbf{3}\) each time, we get:

\[\begin{align} &14 + 3 = 17 \\ &17 + 3 = 20 \\ &20 + 3 = 23 \\ &23 + 3 = 26 \\ \end{align}\]

The number pattern that is formed is {\(14; 17; 20; 23; 26\)}.

Worked example 4.1: Find the next number in the pattern

You are given a pattern that starts with the following numbers:

\[18; 13; 8; 3;\ldots\]

What is the next number in the pattern?

Work out what the pattern is.

We have the numbers, \(18; 13; 8; 3;\ldots\) and we need to find the next number in the pattern.

To do so, we must either work out how the pattern is changing from one term to the next, or notice that the pattern is a special set of numbers.

For this pattern, we can see that to get from one term to the next, we subtract \(5\).

Extend the pattern to get the next number.

To get the next number in the pattern, we must subtract \(5\) from \(3\):

\[3 - 5 = - 2\]

Therefore, the next number in the pattern is \(- 2\).

Worked example 4.2: Find the next number in the pattern

A pattern starts with the following numbers:

\[8; 13; 18; 23;\ldots\]

What is the next number in the pattern?

Work out what the pattern is.

We have the numbers, \(8; 13; 18; 23;\ldots\) and we need to find the next number in the pattern.

To do so, we must either work out how the pattern is changing from one term to the next, or notice that the pattern is a special set of numbers.

For this pattern, we can see that to get from one term to the next, we add \(5\).

Extend the pattern to get the next number.

To get the next number in the pattern, we must add \(5\) to \(23\):

\[23 + 5 = 28\]

Therefore, the next number in the pattern is \(28\).

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Multiplying and dividing by the same number

If we multiply or divide by a number to get the next term in the sequence, then the ratio between the first and second numbers is the same as the ratio between the second and third numbers. Sequences in which the number we multiply or divide by remains the same therefore have a constant ratio.

constant ratio
when the ratio between one number and the next in a sequence is the same as the ratio between any two other consecutive numbers in that sequence

Worked example 4.3: Finding the next number in the pattern

A pattern starts with the following numbers:

\[400; 200; 100; 50;\ldots\]

What is the next number in the pattern?

Work out what the pattern is.

We have the numbers, \(400; 200; 100; 50;\ldots\) and we need to find the next number in the pattern.

To find this, we must either work out how the pattern is changing from one term to the next or notice that the pattern is a special set of numbers.

For this pattern, we can see that to get from one term to the next one we divide by \(2\).

Extend the pattern to get the next number.

To get the next number in the pattern, we must divide \(50\) by \(2\):

\[50 \div 2 = 25\]

Therefore, the next number in the pattern is \(25\).

Worked example 4.4: Finding the next number in the pattern

A pattern starts with the following numbers:

\[3; 9; 27; 81;\ldots\]

What is the next number in the pattern?

Work out what the pattern is.

We have the sequence, \(3; 9; 27; 81;\ldots\) and we need to find the next number in the pattern.

To find it, we must either work out how the pattern is changing from one term to the next or notice that the pattern is a special set of numbers.

For this pattern, we can see that to get from one term to the next one we multiply by \(3\).

Extend the pattern to get the next number.

To get the next number in the pattern, we must multiply \(81\) by \(3\):

\[81 \times 3 = 243\]

Therefore, the next number in the pattern is \(243\).

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Special sequences: fractions and negative numbers

Sometimes the rule to create a pattern does not follow a common difference or constant ratio relationship. Some patterns use fractions or very large negative numbers. You need to look at such patterns carefully, because different rules might apply to them.

For example, this sequence \({ 1; 2; 3; 5; 8;\ldots}\) is made by adding two preceding terms.

\[\begin{align} &1 + 2 = 3 \\ &2 + 3 = 5 \\ &3 + 5 = 8 \\ \end{align}\]

So, the next two terms in this sequence are: \(5 + 8 = 13\) and \(8 + 13 = 21\).

The sequence is \({ 1; 2; 3; 5; 8; 13; 21}\).

If you only looked for a common difference between the first three terms, you would have made a mistake: \(2 - 1 = 1\) and \(3 - 2 = 1\), but \(5 - 3 = 2\)!

When working with fractions or negative numbers, be careful how you do the calculations. It is easy to make a careless mistake and get the incorrect sequence. Look out for square and cube numbers, roots, and other non-linear relationships.

non-linear relationship
a relationship between two variables in which a change in one does not cause a corresponding change in the other

There are a few special, non-linear patterns that you should know and recognise.

In this table, \(n\) stands for each consecutive number, in other words, first you use \(1\) in the formula, then you use \(2\), and then \(3\), and so on.

Name Pattern Formula
Perfect squares \(1;4;9;16;\ldots\) \(t = n^{2}\)
Perfect cubes \(1;8;27;64;\ldots\) \(t = n^{3}\)
Powers of \(2^{}\) \(2;4;8;16;\ldots\) \(t = 2^{n}\)
Powers of \(3^{}\) \(3;9;27;81;\ldots\) \(t = 3^{n}\)
Reciprocals \(1;1^{2};1^{3};1^{4};\ldots\) \(t = 1^{n}\)

Worked example 4.5: Working with fractions in a pattern

Consider the following pattern of numbers:

\[1;\frac{1}{8};\frac{1}{27};\frac{1}{64};\ldots\]

What is the next number in the pattern?

Solve the pattern by inspection.

This pattern is not linear. Look at the table above to see if you can identify the special relationship.

The pattern shown here is made of perfect cubes.

\[1;\frac{1}{\left( 2^{3} \right)};\frac{1}{(3^{3})};\frac{1}{(4^{3})};\ldots\]

Find the next number in the pattern.

Each term in the pattern is the reciprocal of a perfect cube. The next perfect cube is \(5^{3} = 125\), and we must take its reciprocal.

Therefore, the next number in the pattern is \(\frac{1}{125}\).

Worked example 4.6: Working with negative numbers in a pattern

Consider the following pattern of numbers:

\[- 3; - 9; - 27; - 81;\ldots\]

What is the next number in the pattern?

Solve the pattern by inspection.

This pattern is not linear.

The pattern shown here is made of powers of \(3\).

\(- 3 = - 3^{1}\) \(- 9 = - 3^{2}\) \(- 27 = - 3^{3}\) \(- 81 = - 3^{4}\)

Notice how the minus sign is not placed inside the brackets together with the number \(- 3\). This is because all the terms in this sequence are negative, in other words the sign doesn’t change.

Find the next number in the pattern.

Each term in the pattern is the negative of a power of \(3\). The next power of \(3\) is \(3^{5} = 243\), and we must take the negative of it.

Therefore, the next number in the pattern is \(−243\).

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Sometimes working out a pattern is easy, and sometimes it is difficult. This is because there is usually no way to find the pattern by calculating anything. If you do not spot the pattern right away, you must investigate what you see and work out what the pattern is: look at the numbers in the sequence and hunt for a pattern!

This is especially true for working with fractions.

Worked example 4.7: Working with fractions in a sequence

Here is a pattern of fractions:

\[1;\frac{1}{2};\frac{1}{3};\frac{1}{4};\frac{1}{5};\ldots\]

What is the next fraction in the pattern?

To investigate the pattern like this, you can try these steps:

Determine the properties of the items in the pattern.

In this case:

  • the pattern if made of fractions
  • the denominators are the natural numbers.

In this case, focus on the set of natural numbers:

\[0; 1; 2; 3; 4; 5; 6;...\]

Compare the numbers in the question to the “normal” list.

You can see that the denominators increase by \(1\) each time.

\[1;\frac{1}{2};\frac{1}{3};\frac{1}{4};\frac{1}{5};\ldots\]

So, the next number in this sequence is \(\frac{1}{6}\).

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Identify the rule in a pattern

Given a sequence of numbers, we can also identify a pattern or relationship between the term and its position in the sequence. This allows us to predict the value of a term in a sequence based on the position of that term.

It is useful to represent these sequences in tables to help visualise the position of the term. It also makes it easier to describe the general rule for the pattern.

For example, the rule is: “Multiply the position of the number by \(3\) and add \(2\) to the answer.”

We can write this rule as a number sentence:

\[\text{position of the number} \times 3 + 2\]

Use this number sentence to find the terms of the sequence:

\[\begin{align} &1 \times 3 + 2 = 5 \\ &2 \times 3 + 2 = 8 \\ &3 \times 3 + 2 = 11 \\ \end{align}\]

Now draw a table of values to show how the pattern is formed.

Position in sequence 1 2 3 4 $$n$$
Rule:
position of the number  × 3 + 2
1 × 3 + 2 = 5 2 × 3 + 2 = 8 3 × 3 + 2 = 11 4 × 4 + 2 = 18 n × 3 + 2 = 3n + 2

Worked example 4.8: Working with position of a number in a sequence

The table below shows a sequence. However, one of the terms is missing. Fill in the missing value.

Position \(1\) \(2\) \(3\) \(4\) \(5\)
Term \(1\) ? \(9\) \(16\) \(25\)

Write the values in a list.

The table shows a sequence with the second term missing. To simplify the question a bit, we can just look at the sequence in the “normal” way, without a table:

\[1; ?; 9; 16; 25\]

To find the missing term, we must work out what the pattern is: do the terms grow by means of addition, subtraction, multiplication, or division? (Or maybe even some other pattern!)

Work out what the pattern is.

To work out what the pattern is, compare the numbers to each other. Are they getting bigger or smaller? Are the growing slowly or very quickly?

The numbers in this sequence are all square numbers. It is important that you recognise these special numbers:

\[1 = 1^{2}; ?; 9 = 3^{2}; 16 = 4^{2}; 25 = 5^{2}\]

Find the missing number.

We need to fill in the missing square number, which is the square number following \(1\).

\[2^{2} = 4\]

The value of the missing term is: \(4\).

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The rule helps us find the terms in the sequence. We use basic algebra to describe the rules in a “compact” way, without describing them in words. Working with algebraic expressions and variables is an important skill to develop when dealing with sequences.

Worked example 4.9: Identifying terms in a sequence

Consider the following sequence:

\[4a + 4; 6a + 2; 8a; 10a - 2; 12a - 4;\ldots\]

What is the second term in the sequence?

Identify each of the terms to find the answer.

The sequence is a bit crowded with all of those numbers and variables; it will be helpful to organise the terms into a table.

Position \(1\) \(2\) \(3\) \(4\) \(5\)
Term \(4a + 4\) \(6a + 2\) \(8a\) \(10a - 2\) \(12a - 4\)

Locate the term in a sequence.

With the terms organised in the table, it is easier to see that the second term is \(6a + 2\).

Can you find the next term in this sequence? Try subtracting the terms from each other:

\[\begin{align} &(6a + 2) - (4a + 4) = (6a - 4a) + (2 - 4) = 2a - 2 \\ &(8a) - (6a + 2) = (8a - 6a) + (0 - 2) = 2a - 2 \\ &(10a - 2) - (8a) = (10a - 8a) + ( - 2 - 0) = 2a - 2 \\ &(12a - 4) - (10a - 2) = (12a - 10a) + \left( - 4 - ( - 2) \right) = 2a - 2 \\ \end{align}\]

The common difference is \((2a - 2)\). So, the \(6\)th term is then \((12a - 4) + (2a - 2) = (12a + 2a) + ( - 4 - 2) = 14a - 6\).

Worked example 4.10: Evaluating expressions and patterns

This sequence is made of numbers. However, the fourth term is an expression with the variable \(a\).

\[- 8; 0; 8; 8a - 16;\ldots\]

If \(a = 4\), calculate the value of the fourth term.

Substitute into the expression for the missing term.

You have an expression for the fourth term and need to find the value of that term if \(a = 4\). Nothing else in the pattern matters: it is all about the expression \(8a - 16\).

Substitute \(a = 4\) into the expression \(8a - 16\):

\[8(4) - 16\]

Calculate the value of the term.

\[8a - 16 = 8(4) - 16 = 16\]

Therefore, the fourth term is equal to \(16\).

Worked example 4.11: Evaluating expressions and patterns

Determine the value of the fifth term in the same sequence as in Worked example 4.10:

\[-8; 0; 8; 8a - 16;…\]

Identify the pattern in the sequence.

Using the answer for the fourth term, the sequence is now: \(- 8; 0; 8; 16;\ldots\)

Start by working out if the pattern is based on addition, subtraction, multiplication, or division. Be patient, it can take a while to work out what the pattern is!

This sequence has an addition pattern: each term is the sum of the previous term and \(8\).

Work out the value of the next term.

You can organise this information by adding some details to the sequence. For example:

To get the next term, you must work out: \(16 + 8 = 24\).

Therefore, the fifth term is \(24\).

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