1.04e–fPureCore

Recurrence Relations

Q: How is it different from a position-to-term formula?

A position-to-term formula like u_n = 3n + 1 lets you compute any term directly from n. A recurrence only tells you how to get the next term from the current one, so you generally build the sequence in order.

Q: What is a periodic sequence?

One whose terms repeat in a fixed cycle, like 2, 12, 2, 12, (period 2). You can find far-off terms by working out where they fall in the cycle instead of iterating every step.

Q: How do I find the limit of a convergent sequence?

At the limit the terms stop changing, so set u_n+1 = u_n = L and solve for L. This gives the value the sequence approaches.

A recurrence relation defines each term from the one before, rather than from its position. It is how sequences are often generated in real life — this month’s value depends on last month’s — and it introduces periodic and convergent behaviour.

24 min Video by TLMaths Sequences and Series

The big picture

There are two ways to describe a sequence. A $\textbf{position-to-term}$ rule (like $u_n = 3n + 1$ ) lets you jump straight to any term. A $\textbf{recurrence relation}$ (or term-to-term rule) instead says how to get the next term from the current one, such as $u_{n+1} = u_n + 3$ . The recurrence view matches how change actually accumulates — savings that grow by a percentage each year, a drug clearing from the body each dose, a population changing each generation. Learning to generate terms, and to classify a sequence as increasing, decreasing, periodic or convergent, is what turns this from notation into a modelling tool.

A-Level Maths D2-02: Inductive Definitions and Recurrence RelationsCurated by Kepler, not produced by us. Watch it, then work through the notes below.

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What you'll be able to do

Distinguish position-to-term from term-to-term (recurrence) definitions
Generate terms from a recurrence relation and a starting value
Recognise periodic, increasing, decreasing and convergent sequences
Interpret a recurrence in a modelling context

Two ways to define a sequence

A $\textbf{position-to-term}$ rule gives $u_n$ directly in terms of $n$ : put in the position, get the term. A $\textbf{recurrence relation}$ gives $u_{n+1}$ in terms of $u_n$ : it tells you how to step from one term to the next, and needs a starting value to get going.

For example $u_{n+1} = u_n + 3$ with $u_1 = 2$ generates $2, 5, 8, 11, \ldots$ — the same sequence a position rule $u_n = 3n - 1$ describes, but built step by step.

u_{n+1} = f(u_n), \quad u_1 = a

Each term from the previous one, plus a starting value.

Example 1A sequence has

u_{n+1} = 2u_n - 1

with

u_1 = 3

. Find the first four terms.

u_1 = 3

u_2 = 2(3) - 1 = 5

u_3 = 2(5) - 1 = 9

u_4 = 2(9) - 1 = 17

Answer

3, 5, 9, 17

Tip — Keep $u_n$ and $u_{n+1}$ straight: $u_n$ is the term you have, $u_{n+1}$ is the one you are computing. Muddling them is the main pitfall.

Describing long-term behaviour

Once a sequence is generated, you describe how it behaves. It may be $\textbf{increasing}$ (each term larger), $\textbf{decreasing}$ (each smaller), $\textbf{periodic}$ (the terms cycle and repeat with a fixed period), or $\textbf{convergent}$ (the terms settle towards a limit).

A periodic sequence is worth spotting: $u_{n+1} = \tfrac{1}{u_n}$ with $u_1 = 2$ gives $2, \tfrac12, 2, \tfrac12, \ldots$ — period 2. In such cases you can find far-off terms by using the cycle rather than computing every step.

Why this works

A $\textbf{convergent}$ sequence approaches a $\textbf{limit}$ $L$ where the term stops changing: $L = f(L)$ . Setting $u_{n+1} = u_n = L$ and solving is exactly how you find the value an iterative process settles on — the seed of the Numerical Methods chapter.

Recurrences as models

Recurrence relations shine in modelling, because “next depends on now” is how most real processes work. A savings account with interest, a course of medication, or a fish population each naturally give a recurrence, and summing the terms answers questions like “total taken after 10 doses”.

Example 1A patient takes 10 mg of a drug daily. Each day 20% of the drug present is cleared before the next dose. Write a recurrence for the amount

u_n

just after the

n

-th dose.

1Just before the next dose, 80% remains:

0.8 u_n

2Then a new 10 mg dose is added.

3So

u_{n+1} = 0.8 u_n + 10

, with

u_1 = 10

Answer

u_{n+1} = 0.8u_n + 10,\; u_1 = 10

Think like an examiner

Track the subscripts carefully: substitute the current term $u_n$ to produce $u_{n+1}$ , one step at a time.
For a periodic sequence, state the period and use the cycle to find distant terms rather than iterating dozens of times.
To find a limit of a convergent sequence, set $u_{n+1} = u_n = L$ and solve the resulting equation.
In modelling questions, define your variable clearly (what $u_n$ represents and when it is measured) before writing the recurrence.

Common misconceptions

Confusing

u_n

(the current term) with

u_{n+1}

(the next).

Substitute the known

u_n

into the rule to get

u_{n+1}

; keep the indices distinct.

Trying to jump to the 50th term of a recurrence by a position formula that was not given.

Either iterate, or exploit periodicity/convergence — a recurrence does not always have a simple closed form.

Forgetting the starting value when generating terms.

A recurrence needs an initial term (

u_1

) to be defined.

Assuming every sequence converges.

Sequences can increase without bound, oscillate, or cycle; check the behaviour before claiming a limit.

Recurrence essentials

u_{n+1} = f(u_n)

Term-to-term rule.

u_1 = a

Needs a starting value.

L = f(L)

Limit of a convergent sequence.

Stretch yourself

A sequence is defined by $u_{n+1} = \tfrac{1}{2}u_n + 3$ with $u_1 = 4$ . Show that it converges, and find its limit.

Hint — Generate a few terms to see the trend, then set $u_{n+1} = u_n = L$ and solve for the limit.

Questions students ask

What is a recurrence relation?

A rule that gives each term of a sequence from the previous one, such as

u_{n+1} = 2u_n - 1

, together with a starting value. It defines the sequence step by step rather than by position.

How is it different from a position-to-term formula?

A position-to-term formula like

u_n = 3n + 1

lets you compute any term directly from

n

. A recurrence only tells you how to get the next term from the current one, so you generally build the sequence in order.

What is a periodic sequence?

One whose terms repeat in a fixed cycle, like

2, \tfrac12, 2, \tfrac12, \ldots

(period 2). You can find far-off terms by working out where they fall in the cycle instead of iterating every step.

How do I find the limit of a convergent sequence?

At the limit the terms stop changing, so set

u_{n+1} = u_n = L

and solve for

L

. This gives the value the sequence approaches.

Key takeaways

A recurrence defines $u_{n+1}$ from $u_n$ and needs a starting value.
Position-to-term rules jump to any term; recurrences build the sequence in order.
Sequences may be increasing, decreasing, periodic or convergent.
A limit satisfies $L = f(L)$ — the fixed-point idea.
Recurrences model real “next depends on now” processes.

How this fits the course

Sigma Notation

Test yourself

Answer a few questions, then review anything you missed.

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