The first thing to remember about sequences is that there are different types. These include arithmetic, geometric, quadratic, and linear sequences. They are just a list of numbers (or other objects) that follow a rule. The key is to identify this rule and work out the nth term, which is a formula to find any term in the sequence.
This article will explore the sequence types you're expected to know for GCSE Maths. We will explain term to term rules and position-to-term rules. Quiz questions are included to test your understanding. This guide is suitable for all major exam boards, including AQA and Edexcel.
If you are a student and need help to learn sequences, TeachTutti has qualified GCSE Maths tutors who can help you revise this topic. For example, preparing revision notes and walking through exam questions and worksheets.
Types of sequences
Simply put, a sequence is a list of numbers that follow a specific pattern or rule. Every number in the sequence is called a term:
- The term increments by 2 | 2, 4, 6, 8, etc
- The term multiples by 3 | 3, 9, 27, 81, etc
- Every term is a square number | 1, 4, 9, 16, etc
It doesn't have to be a number. A sequence could be of geometric objects, such as squares growing in size. It could also be letters or symbols, such as a sequence of colours (Red, Blue, Green, etc).
These are the sequence types you are expected to know for GCSE Maths:
- Arithmetic sequences | The same number is added/subtracted for each term, e.g. 2, 5, 8, 11, etc (3 is added each time)
- Geometric sequences | The same number is multiplied/divided for each term, e.g. 3, 6, 12, 24, etc (3 is multiplied each time)
- Quadratic sequences | The difference between terms changes in a regular way. This is often linked to square numbers (multiplying a number by itself), e.g. 1, 4, 9, 16, etc
- Practical sequences | Sequences that arise from real-life situations, e.g. the number of steps in a staircase
1
Which is an arithmetic sequence?
Term-to-Term rules
A term-to-term rule in a sequence explains how to move from one term to the next. If we know this rule and the starting term, we can calculate as many terms as we want.
For example:
- 3, 7, 11, 15, etc | 4 is added between each term. This means the term-to-term rule is to start with 3 and add 4 each time
- -1, -0.5, 0, 0.5, etc | 0.5 is added each time. The term-to-term rule is to start with -1 and add 0.5 each time
- 1, 2, 4, 8, 16, etc | The term is multiplied by 2 each time. The rule is to start with 1 and multiply each term by 2
To find the correct rule, always start by checking if the change is a constant (addition/subtraction) or a ratio (multiplication/division).
2
What is the rule for the sequence 17, 14, 11, 8, etc?
Position-to-term rules | Find the nth term
A position-to-term rule is an algebraic expression that lets you find any term in a sequence. Using this algebraic formula, you don't need to work through each previous term. It is normally written in the form of an nth term.
For example, find the nth term of linear sequence 5, 9, 13, 17, etc:
- The nth term is 4n + 1
- This means the difference between each algebraic term is +4
- The first term (n = 1) is 4(1) + 1 = 5
- The second term (n = 2) is 4(2) + 1 = 9
- The third term (n = 3) is 4(3) + 1 - 13
This rule is particularly useful when it is a larger nth term, as you don't need to write out each term. For instance, trying to find the 50th term in a sequence.
When using a geometric sequence, we often write the nth term as a x r^(n-1). This means a is the first term and r is the common ratio.
For example, find the nth term for the simple geometric progression 5, 10, 20, 40, etc:
- The nth term is therefore 5 × 2^(n-1)
- This is because the first term (a) is 5 and the common ratio (r) is 2. This means each term is multiplied by 2
- The first term (n = 1) is 5 × 2^0 = 5
- The second term (n = 2) is 5 × 2^1 = 10
- The fourth term (n = 3) is 5 × 2^2 = 20
3
What is the nth term of 2, 6, 10, 14, etc?
Arithmetic sequences
An arithmetic sequence is when each term increases or decreases by the same amount. This amount is called the common difference. For example, in the simple arithmetic progression 3, 7, 11, 15, etc, the common difference is +4. This means the nth term is 4n - 1.
You can turn the position-to-term rule around and use the nth term to find where a term belongs in a sequence. For example, where does 35 belong in 3, 7, 11, 15, etc:
- The nth term is 4n - 1
- We need to solve 4n - 1 = 35. 4n = 36, which means n is 9 (36 divided by 4)
- This means 35 is the 9th term. It fits into the sequence because it is a whole number
Remember that if your calculations for the nth term don't return a whole number, this means the number given is not part of the sequence.
4
Which number is in the sequence 2, 5, 8, 11, etc?
Geometric Sequences
A geometric sequence is when you multiply or divide each term by the same number. This number is called the common ratio. In other words, every term is linked to the previous and next by multiplication.
For example, 2, 6, 18, 54, etc:
- The ratio for each term is ×3
- If you divide any term by the previous term, you always get 3. For instance, 18 / 6 = 3 and 6 / 2 = 3. This confirms that the sequence is geometric
We write the nth term in a geometric sequence as a x r^(n-1), where "a" is the first term in the sequence and "r" is the common ratio.
Example 1
Find the 5th term of 3, 6, 12, 24, etc:
- The first term (a) is 3
- The common ratio (r) is 2. This is because 6 / 3 = 2 ( and 12 / 6 is 2, and so on)
- This means the nth term is 3 x 2^(n-1)
- The 5th term is 48 (3 x 2^4 = 3 x 16 = 48)
Example 2
Find the 7th term of 5, 15, 45, 135, etc:
- The first term (a) is 5
- The common ratio (r) is 3 (15 / 5 = 30
- The nth term is 5 × 3^(n-1)
- The 7th term is 3645 (5 × 3^6 = 5 × 729 = 3645
5
What is the nth term of 4, 8, 16, 32, etc?
Quadratic sequences
When the difference between terms isn't constant, the sequence isn't arithmetic. Instead, if the difference of the differences is constant, this is a quadratic sequence. The difference of the differences is known as the second difference.
Quadratic sequences are linked to square numbers. The formula for the nth term is an² + bn + c. The letters a, b and c are the numbers you need to find.
Start by checking if the sequence is quadratic. For example, 2, 6, 12, 20, 30, etc:
- The first differences between terms are 4, 6, 8, 10
- The second differences (the difference between these differences) is 2, 2, 2
- The second difference is constant, which means this is a quadratic sequence
Next, find the value of a:
- We find a when we divide the second difference by 2
- This means a = 1 (2 / 2 = 1)
- The nth term starts with n²
Now we need to calculate b and c:
- The formula is now n² + bn + c
- Find the first term (n = 1) | 2 -> 1² + b(1) + c = 2 -> 1 + b + c = 2 -> b + c = 1
- To find the second term (n = 2) | 6 -> 2² + b(2) + c = 6 -> 4 + 2b + c = 6 -> 2b + c = 2
- The first equation is b + c = 1 and the second equation is 2b + c = 2
- We now subtract them | (2b + c) - (b + c) = 2 – 1 -> b = 1
- This means c = 0
- Therefore, the nth term is n² + n
Finally, check your answers to ensure you haven't made a mistake in your workings:
- The first term (n = 1) is 2 | 1² + 1 = 2
- The second term (n = 2) is 6 | 2² + 2 = 6
- The third term (n = 3) is 12 | 3² + 3 = 12
6
What is the nth term for 1, 4, 9, 16, etc?
Special sequences
There are special sequences that don't relate to the arithmetic, geometric, or quadratic categories. These sequences follow unique rules. The most famous example is the Fibonacci sequence.
Fibonacci numbers
The Fibonacci sequence starts with 0 and 1. Every new term is then the sum of the previous two terms. For example: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc.
There are interesting properties to these sequences. For instance, the results get closer to approximately 1.618 if you divide each term by the one before it. This is called the golden ratio. It is used in art and architecture.
You will also find Fibonacci numbers in the properties of living things. The arrangement of petals on a flower, the pattern of a pine cone, and the spiral of a sunflower all follow Fibonacci rules. This sequence is therefore both mathematically and deeply connected to the world we live in.
These are other examples of special sequences:
- Square numbers | These are numbers multiplied by themselves. For example, the sequence 1, 4, 9, 16 is created by 1 x 1, 2 x 2, 3 x 3, and so on
- Triangular numbers | These are created by adding consecutive natural numbers. For example, the sequence 1, 3, 6, 10, 15 is created by 1, 1 + 2, 1 + 2 + 3, and so on
- Cube numbers | These are integers created by multiplying another number by itself twice. For example, the sequence 1, 8, 27, 64 is created by 1 x 1 x 1, 2 x 2 x 2, 3 x 3 x 3, and so on
- Odd numbers | 1, 3, 5, 7, etc
- Even numbers | 2, 4, 6, 8, etc
7
What is the next term in the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8?
Conclusion
Sequences are seen throughout maths, from simple number patterns to more complex quadratic and geometric progressions. We have covered the main types of sequences and briefly explored special types, including the Fibonacci sequence.
When approaching a question on the topic, always remember to begin by identifying the correct type of sequence. You can then apply the relevant formula, such as finding the common difference between the terms, finding a ratio, or calculating the nth term. As always, the key to mastering the different types of sequences is practice, including avoiding common mistakes in your exams.
A related link we suggest is on a past paper on sequences by MathsGenie. You can also learn about geometric sequences in more detail with this article from ThirdSpaceLearning.
If you’d like personalised support, TeachTutti has top GCSE Maths tutors who can teach you in person or online using the TeachTutti learning platform. Every tutor has an enhanced DBS check before they can begin tuition.
