*Arithmetic Sequences with a common difference may be defined by a first order difference equation.*

## The Equation

An arithmetic sequence where there is a common difference of can be defined by a first order difference equation of the form:

Where is the common difference

is an increasing sequence

is a decreasing sequence

## First Order Difference Equations Defining Arithmetic Sequences

A difference equation must have a common difference and no common ratio for it to define an arithmetic sequence.

### Example 1

**Determine whether each of the following first order difference equations defines and arithmetic sequence.**

**a) **

The first order difference equation defines a decreasing arithmetic sequence with a common difference of -7.

**b) **

The first order difference equation defines an increasing arithmetic sequence with a common difference of 5.

**c) **

The first order difference equation does *not* define an arithmetic sequence as the term has a coefficient of 3.

## Turning Arithmetic Sequences into First Order Difference Equations

An arithmetic sequence can be expressed as a first order difference equation by finding the initial or starting term and the common difference. This is then substituted into the general equation above.

### Example 2

**Express each of the following arithmetic sequences as first order difference equations.**

**a) **

First we must check for a common difference.

We know that the first term is -22 and the common difference is 7 so we can now put these into the equation.

**b) **

Again, we must check for a common difference.

We know that the first term is 23 and the common difference is -5 so we can now put these into the equation.

#### See also:

Recognition of Arithmetic Sequences

Finding the Terms of an Arithmetic Sequence

First Order Difference Equations

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

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