3.7PureStretch

Recurrence Relations

A recurrence relation defines each term from the previous one, like $u_{n + 1} = 2 u_{n} + 3$ . It generates sequences step by step, and the language of increasing, decreasing and periodic sequences describes their behaviour.

25 min Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.7 Recurrence RelationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Generate terms from a recurrence relation
Understand the need for a starting term
Classify increasing, decreasing and periodic sequences
Recognise periodic sequences

Generating terms

A recurrence relation such as $u_{n + 1} = f (u_{n})$ gives the next term from the current one. You need a $starting value$ $u_{1}$ to begin, then apply the rule repeatedly.

u_{n + 1} = f (u_{n}), u_{1} given

Each term comes from the one before.

u_{2} = 2 (3) + 1 = 7

u_{3} = 2 (7) + 1 = 15

Answer

u_{3} = 15

Increasing, decreasing, periodic

A sequence is $increasing$ if $u_{n + 1} > u_{n}$ for all $n$ , $decreasing$ if $u_{n + 1} < u_{n}$ for all $n$ , and $periodic$ if the terms repeat in a fixed cycle.

Periodic sequences

A periodic sequence has $u_{n + k} = u_{n}$ for some fixed period $k$ — the values cycle. For example $u_{n + 1} = \frac{1}{u _{n}}$ starting at $2$ gives $2, \frac{1}{2}, 2, \frac{1}{2}, \dots$ (period 2).

Tip — To spot a period, generate several terms and look for the cycle length k where they start repeating.

Formula recap

u_{n + 1} = f (u_{n})

Recurrence relation.

u_{n + 1} > u_{n} \forall n \Rightarrow increasing

Increasing.

u_{n + k} = u_{n} \Rightarrow periodic (period k)

Periodic.

Common mistakes to avoid

Trying to generate terms without a starting value.

A recurrence needs an initial term (e.g. u₁).

Calling a sequence periodic after one repeat by coincidence.

Periodic means the cycle repeats consistently with a fixed period.

Key takeaways

Recurrence: uₙ₊₁ = f(uₙ), with a given starting term.
Increasing/decreasing: every step rises/falls.
Periodic: terms repeat with a fixed period k.

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