Worked example 1
Identify a pattern
Classify 3, 7, 11, 15, ...
- Compare consecutive terms.
- The difference is +4 each time.
Answer: Arithmetic sequence with common difference 4.
Higher Mathematics
Higher Maths Sequences and limits
Practise sequences and limits for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Review sequence notation and use successive values to support recurrence-limit reasoning.
Before you start
- Recognise arithmetic and geometric patterns.
- Use term notation such as uₙ.
- Compare successive terms to see a trend.
Explanation
Sequences are ordered lists of terms. Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio.
A sequence may approach a limit. At Higher, this is often connected to repeated recurrence values or long-term behaviour.
Method and rules
- Arithmetic: uₙ = a + (n − 1)d
- Geometric: uₙ = arⁿ⁻¹
- A convergent sequence approaches a limiting value.
Worked examples
Worked example 2
Use a geometric rule
Find the 5th term of 2, 6, 18, ...
- The first term is 2 and ratio is 3.
- Use uₙ = arⁿ⁻¹ with n = 5.
So: u₅ = 2×3⁴ = 162
Watch out
- Using n instead of n − 1 in nth-term formulae.
- Mixing up common difference and common ratio.
- Rounding repeated values too early.
- Assuming every increasing sequence has a finite limit.
Exam reminder
State whether the pattern is arithmetic, geometric or recurrence-based before applying a formula.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.