Higher Mathematics

Higher Maths Recurrence modelling and limits

Practise recurrence modelling and limits for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Derive linear recurrence models, find limits where they exist and interpret them in context.

Topic practiceFormula supportMixed revision

Before you start

  • Calculate successive recurrence terms.
  • Convert percentages to multipliers.
  • Solve linear equations.
Higher Mathematics lesson

Explanation

A context often becomes uₙ₊₁ = auₙ + b: first apply the retained multiplier a, then add or subtract the fixed change b.

If |a| < 1, a limit may exist. Substitute L for consecutive terms and interpret the result using the model's units and constraints.

Method and rules

  • Linear recurrence: uₙ₊₁ = auₙ + b
  • Candidate limit: L = aL + b, so L = when a is not 1
  • For this linear model, |a| < 1 gives convergence to the fixed point.

Worked examples

Worked example 1

Model and interpret a limit

A tank retains 80% of its contents each cycle and then receives 12 litres. Write a recurrence relation and find its limiting amount.

  1. Use retained amount 0.8uₙ.
  2. Add 12 to obtain uₙ₊₁ = 0.8uₙ + 12
  3. Solve L = 0.8L + 12.

Answer: uₙ₊₁ = 0.8uₙ + 12 and the limit is 60 litres

Worked example 2

Method check

Find the limit of uₙ₊₁ = 0.5uₙ + 7, where it exists.

  1. Solve L = 0.5L + 7.

So: 14.

Watch out

  • Using 0.2 instead of the retained multiplier 0.8.
  • Finding a fixed point without checking convergence.
  • Giving a limit without units or context.

Exam reminder

For recurrence modelling and limits, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.

Related Higher Maths topics

Continue with the methods that connect most closely to this topic.

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