Use first-order linear recurrence relations to generate sequences and recognise the patterns of growth and decay they produce.

How to read and use a first-order linear recurrence relation, generate terms step by step, and recognise when it produces linear, growing or decaying behaviour.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
Anatomy of a recurrence relation
Generating terms
Why recurrence relations matter

What this dot point is asking

You must read a recurrence relation, identify the starting value and the rule, generate successive terms, and recognise the type of behaviour produced.

Anatomy of a recurrence relation

A recurrence relation defines a sequence using two parts: a starting value and a rule linking each term to the previous one.

The subscript $n$ counts the steps. The starting value may be written $t_0$ or $t_1$ depending on the question, so read carefully which term the count begins from.

Generating terms

To generate the sequence, substitute the current term into the rule to get the next, then repeat. This step-by-step process is the heart of every growth and decay model in the course.

Worked example

Generating a sequence from a recurrence

A sequence is defined by $t_{n+1} = 0.8\,t_n + 50$ with $t_0 = 200$ . Find the first four terms.

Start: $t_0 = 200$ .

$t_1 = 0.8 \times 200 + 50 = 160 + 50 = 210$ .

$t_2 = 0.8 \times 210 + 50 = 168 + 50 = 218$ .

$t_3 = 0.8 \times 218 + 50 = 174.4 + 50 = 224.4$ .

The terms are $200, 210, 218, 224.4, \dots$ They rise but by smaller amounts each step, settling towards a long-run value. This mixed multiply-and-add rule is exactly the structure that models a reducing-balance loan with regular repayments or a population with immigration.

Why recurrence relations matter

Recurrence relations are the unifying idea behind the whole growth and decay topic. Arithmetic sequences, geometric sequences, compound interest, depreciation, loans and annuities are all special cases of the first-order linear recurrence, differing only in the values of $a$ and $b$ . Mastering the step-by-step generation makes every later model routine.