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3.7PureStretch

Recurrence Relations

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

25 min Sequences and Series
Edexcel A level Maths: 3.7 Recurrence RelationsCurated by Kepler, not produced by us. Watch it 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
1

Generating terms

A recurrence relation such as un+1=f(un)u_{n+1} = f(u_n) gives the next term from the current one. You need a starting value\textbf{starting value} u1u_1 to begin, then apply the rule repeatedly.

un+1=f(un),u1 givenu_{n+1} = f(u_n), \quad u_1 \text{ given}
Each term comes from the one before.
Example 1For un+1=2un+1u_{n+1} = 2u_n + 1 with u1=3u_1 = 3, find u3u_3.
1u2=2(3)+1=7u_2 = 2(3) + 1 = 7.
2u3=2(7)+1=15u_3 = 2(7) + 1 = 15.
Answer
u3=15u_3 = 15
2

Increasing, decreasing, periodic

A sequence is increasing\textbf{increasing} if un+1>unu_{n+1} > u_n for all nn, decreasing\textbf{decreasing} if un+1<unu_{n+1} < u_n for all nn, and periodic\textbf{periodic} if the terms repeat in a fixed cycle.

3

Periodic sequences

A periodic sequence has un+k=unu_{n+k} = u_n for some fixed period kk — the values cycle. For example un+1=1unu_{n+1} = \dfrac{1}{u_n} starting at 22 gives 2,12,2,12,…2, \tfrac12, 2, \tfrac12, \ldots (period 2).

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

Formula recap

un+1=f(un)u_{n+1} = f(u_n)
Recurrence relation.
un+1>un ∀n⇒increasingu_{n+1} > u_n \ \forall n \Rightarrow \text{increasing}
Increasing.
un+k=un⇒periodic (period k)u_{n+k} = u_n \Rightarrow \text{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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A recurrence relation defines each term from:

    • The previous term(s)
    • Its position only
    • A fixed sum
    • The last term only
    Show answer and explanation

    Answer: The previous term(s)

    uₙ₊₁ depends on uₙ.

  2. To use a recurrence you need:

    • A starting value
    • The sum
    • The common ratio
    • A graph
    Show answer and explanation

    Answer: A starting value

    e.g. u₁.

  3. For un+1=2un+1u_{n+1} = 2u_n + 1, u1=3u_1 = 3, then u2=u_2 =

    • 77
    • 66
    • 55
    • 1515
    Show answer and explanation

    Answer: 77

    2(3) + 1.

  4. For the same recurrence, u3=u_3 =

    • 1515
    • 1313
    • 1111
    • 99
    Show answer and explanation

    Answer: 1515

    2(7) + 1.

  5. A sequence whose terms repeat in a cycle is:

    • Periodic
    • Increasing
    • Decreasing
    • Convergent
    Show answer and explanation

    Answer: Periodic

    Fixed period.

  6. un+1=1unu_{n+1} = \tfrac{1}{u_n} with u1=2u_1 = 2 gives a sequence of period:

    • 22
    • 11
    • 33
    • 44
    Show answer and explanation

    Answer: 22

    2, ½, 2, ½, … repeats every 2.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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