Special Sequences
A sequence is a list of numbers that follows a specific pattern or rule. Understanding common sequences like triangular, square, and Fibonacci helps develop logical thinking and is a key skill for your GCSE Maths exam.
💻 Programming
Algorithms often use sequences to perform tasks like sorting data or generating graphics.
💰 Finance
Calculations for compound interest or loan repayments rely on the principles of sequences.
🎨 Art & Design
Patterns, from simple tessellations to complex fractals, are all based on mathematical sequences.
Key Sequences to Know
Arithmetic Sequences
The difference between consecutive terms is constant — this is called the common difference.
5 → 8 → 11 → 14 …
(Rule: add 3 each time)
Fibonacci-type Sequences
Each new term is found by adding the two terms before it.
2, 5 → 7 → 12 → 19 …
($2+5=7$, then $5+7=12$, etc.)
Square Numbers
Numbers formed by multiplying an integer by itself ($n^2$).
1, 4, 9, 16, 25…
Cube Numbers
Numbers formed by multiplying an integer by itself three times ($n^3$).
1, 8, 27, 64…
Triangular Numbers
Numbers that can be arranged into a triangular dot pattern.
1, 3, 6, 10, 15…
Worked Example: Identifying Sequences
For each sequence, identify its type and find the next two terms.
1, 4, 9, 16, …
These are $1^2, 2^2, 3^2, 4^2$.
Type: Square Numbers.
Next: $5^2=25$, $6^2=36$.
5, 10, 15, 20, …
Each term increases by +5.
Type: Arithmetic (common difference 5).
Next: 25, 30.
3, 4, 7, 11, …
$3+4=7$, then $4+7=11$.
Type: Fibonacci-type.
Next: $7+11=18$, $11+18=29$.
Tutor Insights
🤔 Common Misunderstandings
- Fibonacci misstep: Adding the first two terms correctly, but then repeating that same sum rather than always using the two immediately preceding terms.
- Not checking the whole sequence: Spotting a difference between the first two terms and assuming it’s arithmetic without checking whether the pattern holds throughout.
📝 Common Exam Mistakes
- Calculation errors, especially with larger cube numbers or negative differences.
- Misidentifying the sequence type. Check for a common difference first, then Fibonacci behaviour, then compare with the special lists.
- Giving too few terms when asked for “the next three terms”.
Practice Questions
- Find the next three terms in the sequence: 20, 18, 16, 14, …
- Identify the type of sequence and explain how you know: 1, 8, 27, 64, …
- A pattern is made with dots: Pattern 1 has 1 dot, Pattern 2 has 3, Pattern 3 has 6. How many dots are in Pattern 5? What type of sequence is this?
- A sequence starts with 4 and 7. Each term is the sum of the two preceding terms. Write the first five terms.
Show Answers
- Rule: subtract 2 each time. Next three terms: 12, 10, 8.
- Type: Cube numbers. Each term is an integer cubed: $1^3, 2^3, 3^3, 4^3$.
- Sequence: 1, 3, 6, 10, 15. Pattern 5 has 15 dots. This is a triangular number sequence.
- 4, 7, 11, 18, 29.
Need Help with Sequences?
A qualified GCSE maths tutor can guide you through sequences, nth term rules, and every other algebra topic on the syllabus.
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