Understanding Sequences
Sequences are lists of numbers that follow a specific pattern or rule. Understanding them helps develop logical thinking and is key for solving problems in your GCSE Maths exam and in everyday life.
💰 Saving Money
If you start with £5 and add £2 every week, you’re creating a sequence to track your savings.
🧱 Building a Tower
A tower with 3 blocks on the first layer, 5 on the second, and 7 on the third follows a sequence.
🏟️ Seating Plans
A concert venue with 10 seats in the first row, 12 in the second, and 14 in the third uses a sequence to manage capacity.
Two Types of Sequence Rules
1. Term-to-Term Rule
This rule tells you how to get from one term to the very next term. It’s like giving step-by-step instructions.
Example: “Start at 5, then add 3 each time.”
5 → 8 → 11 → 14 …
2. Position-to-Term Rule (nth Term)
This is a powerful formula that lets you find any term in the sequence if you know its position ($n$).
Example: The rule is $2n + 1$.
- 10th term ($n=10$): $2(10) + 1 = 21$
- 50th term ($n=50$): $2(50) + 1 = 101$
Worked Examples
Using a Term-to-Term Rule
A sequence starts with 10. The rule is “add 4”. Write the first five terms.
- 1st term: 10
- 2nd term: $10 + 4 = 14$
- 3rd term: $14 + 4 = 18$
- 4th term: $18 + 4 = 22$
- 5th term: $22 + 4 = 26$
Answer: 10, 14, 18, 22, 26
Using an nth Term Rule
A sequence has the rule $3n – 2$. Find the 1st and 20th terms.
- 1st term (n=1):
$3(1) – 2 = 1$ - 20th term (n=20):
$3(20) – 2 = 58$
Answers: 1 and 58
Tutor Insights
🤔 Common Misunderstandings
- Term-to-term vs. nth term: Confusing the two rules. Remember: term-to-term is for the *next* step, while the nth term lets you jump to *any* step.
- What ‘n’ means: Forgetting that $n$ stands for the term’s position ($n=1$ for the first term, $n=2$ for the second, etc.).
📝 Common Exam Mistakes
- Arithmetic errors: A small slip-up in an early term can make all subsequent terms wrong.
- Incorrect substitution: Forgetting that $3n$ means $3 \times n$, not $3+n$.
- Not writing enough terms if the question asks for a specific number.
Practice Questions
- A sequence starts with 7 and has a term-to-term rule of “add 5”. Write down the first five terms.
- A sequence has the nth term rule $n+8$. Find the 15th term.
- A sequence starts with 30 and has a term-to-term rule of “subtract 3”. Write down the first five terms.
- A sequence has the nth term rule $4n-3$. Find the 50th term.
- A sequence begins 7, 12, 17, 22, … What is its nth term rule?
Show Answers
- Answer: 7, 12, 17, 22, 27
- Working: $15 + 8 = 23$.
Answer: 23 - Answer: 30, 27, 24, 21, 18
- Working: $4(50) – 3 = 200 – 3 = 197$.
Answer: 197 - Answer: $5n+2$
FAQs
What’s the main difference between a term-to-term rule and an nth term rule?
A term-to-term rule tells you how to get from one term to the next (e.g., “add 3”). An nth term rule is a formula that lets you find any term in the sequence if you know its position, $n$ (e.g., $3n-2$).
Do I always start with $n=1$ for the nth term rule?
Yes, for GCSE maths, $n=1$ always represents the position of the first term in a sequence.
