Recurrence Relations

A recurrence relation defines each term of a sequence using the previous term(s). Instead of giving an explicit formula for u(n) in terms of n, we give a rule connecting u(n+1) to u(n), along with a starting value. This is an important topic in the Edexcel A-Level (9MA0) specification.

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Key Definitions

Term	Meaning
Recurrence relation	A formula that gives u(n+1) in terms of u(n) (and possibly n)
Initial condition	The starting value, e.g. u(1) = 3, needed to generate the sequence
Increasing sequence	u(n+1) > u(n) for all n
Decreasing sequence	u(n+1) < u(n) for all n
Periodic sequence	A sequence that repeats with a fixed period: u(n+k) = u(n) for all n

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Defining Sequences with Recurrence Relations

A recurrence relation has two parts:

1. The recursive rule, e.g. u(n+1) = 3u(n) - 1
2. The initial condition, e.g. u(1) = 2

Both are needed to uniquely define a sequence.

Worked Example 1

A sequence is defined by u(n+1) = 2u(n) + 3, u(1) = 1. Find the first five terms.