*A first order difference equation defines a relationship between two successive terms of a sequence.*

## The Equation

A difference equation has two parts, the rule describing the pattern between two terms and the first or initial term.

The th term in a sequence is . Therefore, denotes the next term in the sequence, so the (n+1)$th term. Alternatively, can denote the previous term and the next term.

(where

is constant)

The initial term can be denoted by or depending on whether the question provides the first term () or the initial term ().

**Example 1**

**Which of the following equations are first order difference equations?**

**a) **

The equation describes the relationship between two consecutive terms ( and and has a starting term of 4.

Therefore this *is *a first order difference equation.

**b) **

The equation describes the relationship between two consecutive terms ( and ) but does not provide a starting term.

Therefore this is *not* a first order difference equation.

**c) **

The equation provides the initial term but does not describe the relationship between two *consecutive *terms.

Therefore this is *not* a first order difference equation.

#### See also

Generating the Terms of a Sequence Defined by a First Order Difference Equation

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