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.
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Higher Mathematics
Sequences and limits
Use sequence notation, compare terms and discuss convergence from repeated values.
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.