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Higher Maths · Applications

Recurrence Relations | Higher Maths

Recurrence Relations are a small but reliable mark-grab on Higher Maths. Most exams include a 5–7 mark question asking you to find a limit, model a real-world situation (drug doses, fish stocks, savings) or compare two relations.

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SQA Higher MathsSpecification: ApplicationsUnit 1 (legacy)

Linear recurrence relation

General form: un+1 = a·un + b
A limit exists if and only if −1 < a < 1 (i.e. |a| < 1).
Limit formula: L = b / (1 − a)

Worked example

Worked example — Find a limit

Problem: A drug is administered such that 25% remains in the body each hour and 20 mg is given each hour. Write down the recurrence relation and find the long-term level of drug in the body.

  1. Identify a and b.
    25% remains ⇒ a = 0.25; constant dose b = 20. So un+1 = 0.25 un + 20.
  2. Check the condition for a limit.
    |0.25| < 1, so a limit exists.
  3. Apply the limit formula.
    L = 20 / (1 − 0.25) = 20 / 0.75 = 26.67 mg (to 2 d.p.).

Practice questions

Try these SQA-style questions. Tap "Show answer" to check your working.

Practice questions

  1. A sequence is defined by un+1 = 0.6un + 8 with u0 = 5. Find u3.
    Show answer
    u1 = 11, u2 = 14.6, u3 = 16.76
  2. Show that the recurrence relation un+1 = 1.2un + 4 has no limit.
    Show answer
    a = 1.2, |a| = 1.2 > 1, so the condition |a| < 1 fails. No limit exists.
  3. Find the limit of un+1 = 0.7un + 6.
    Show answer
    L = 6 / (1 − 0.7) = 6 / 0.3 = 20
  4. Two recurrence relations have limits 50 and 30. Which is larger? Explain in one sentence.
    Show answer
    50 > 30. The limit indicates the long-term value the sequence approaches.
  5. A savings account has 3% interest each year and an annual deposit of £1500. Write the recurrence relation.
    Show answer
    un+1 = 1.03 un + 1500 (no limit, since |a| > 1)

Common mistakes

Common mistakes & how to avoid them

  • Using the limit formula when |a| ≥ 1 — the limit does not exist in that case.
  • Confusing the multiplier a with the percentage retained or removed — read the question carefully.
  • Forgetting to state the condition |a| < 1 in the working — SQA requires the justification for the mark.

Frequently asked questions

What is a "limit" in this context?
The limit is the value that the sequence approaches as n becomes very large. Once the sequence stabilises, un+1 ≈ un ≈ L.
Will a recurrence relation always have a limit?
No. Only when the multiplier a satisfies −1 < a < 1. If |a| ≥ 1 the sequence either grows without bound or oscillates.
Are recurrence relations on Paper 1 or Paper 2?
They can appear on either paper, but more commonly on Paper 2 because of the calculator-based limit calculations.

Related Higher Maths topics

These topics often appear together in SQA exam questions.

Functions & GraphsLogs & ExponentialsQuadratic Theory

← All Higher Maths topics

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