Counting and sequences
🎯 In this topic you will
- Count on and count back in steps of constant size, including counting back through zero using negative numbers.
- Determine the jump size in a linear sequence and relate it to multiplication.
🧠 Key Words
- linear sequence
- sequence
- term
- term-to-term
Show Definitions
- linear sequence: A number pattern where the difference between consecutive terms is always the same.
- sequence: A list of numbers written in a specific order that follows a clear rule or pattern.
- term: A single value or number within a sequence.
- term-to-term: A rule that shows how to move from one term in a sequence to the next (for example, “add 4 each time”).
🌟 Flash Patterns at a Lighthouse
A lighthouse displays a single flash repeated at regular times.
💡 Multiple Flashes in a Group
It can also display groups of two, three or four flashes with spaces between the groups.
🔁 A Repeating Light Pattern
The whole pattern is repeated at regular intervals.
📘 Learning About Sequences
In this section you will learn about sequences that follow particular rules.
❓ EXERCISES
1. a. Here is a sequence made of sticks. Draw the next pattern in the sequence.
b. Copy and complete the table.
| Pattern number | Number of sticks |
|---|---|
| $1$ | |
| $2$ | |
| $3$ | |
| $4$ |
c. Find the term-to-term rule.
5. Sofia makes a number sequence.
The first term is $155$ and the term-to-term rule is ‘subtract $7$’.
Sofia says, ‘If I keep subtracting $7$ from $155$, I will eventually reach $0$.’
Is she correct? Explain your answer.
👀 Show answer
The $n$th term of the sequence is $155 - 7(n-1)$.
To reach $0$ we would need $155 - 7(n-1) = 0$.
This gives $7(n-1) = 155$, so $n-1 = \dfrac{155}{7}$, which is not an integer (it is about $22.14$).
That means no whole-number term of the sequence is equal to $0$; the terms jump from a positive number to a negative number without ever being exactly $0$.
Sofia is not correct – she will never reach $0$ by repeatedly subtracting $7$ from $155$.
6. The numbers in this sequence increase by equal amounts each time.
Copy the sequence and write in the missing numbers.
$1,\; \square,\; \square,\; 7$
Explain your method.
Discuss your answer with your partner. Did you use the same method?
👀 Show answer
This is an arithmetic sequence with four terms: first term $1$ and last term $7$.
Let the common difference be $d$. Then the terms are:
$1,\; 1+d,\; 1+2d,\; 1+3d = 7$.
So $1 + 3d = 7 \Rightarrow 3d = 6 \Rightarrow d = 2$.
The sequence is therefore $1, 3, 5, 7$.
The missing numbers are $3$ and $5$.
7. Marcus writes a sequence of numbers.
His rule is to add the same amount each time.
Copy the sequence and write in the missing numbers.
$1,\; \square,\; \square,\; \square,\; 21$
👀 Show answer
This sequence has five terms: first term $1$ and last term $21$ with constant difference $d$.
The terms are $1,\; 1+d,\; 1+2d,\; 1+3d,\; 1+4d = 21$.
So $1 + 4d = 21 \Rightarrow 4d = 20 \Rightarrow d = 5$.
The sequence is $1, 6, 11, 16, 21$.
The missing numbers are $6$, $11$ and $16$.
🧠 Think like a Mathematician
You can write a sequence with a constant jump size if you know three pieces of information:
- the first term
- the jump size
- the number of terms in the sequence
This information is given in the table for five sequences. Write each of the sequences A to E.
| First term (number) | Jump size | Number of terms in the sequence |
| A: 1 | +4 | 5 |
| B: 20 | −3 | 5 |
| C: −15 | +11 | 5 |
| D: 100 | −26 | 5 |
| E: −40 | +15 | 5 |
Task: Work out all five sequences by applying the jump size to the first term until all five terms are completed.
After finishing, make up some linear sequences of your own and write down the three pieces of information that define your sequences.
Show Answers
- Sequence A: 1, 5, 9, 13, 17
- Sequence B: 20, 17, 14, 11, 8
- Sequence C: −15, −4, 7, 18, 29
- Sequence D: 100, 74, 48, 22, −4
- Sequence E: −40, −25, −10, 5, 20
Tip: Each sequence follows the rule $\text{next term} = \text{previous term} + \text{jump size}$.