The Relationship Between Arithmetic Sequences and First Order Difference Equations

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 $b$ can be defined by a first order difference equation of the form:

$t_{n+1}=t_n+b$

Where $b$ is the common difference
$b>0$ is an increasing sequence
$b<0$ 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) $t_{n+1}=t_n-7$ $t_1=22$

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

b) $t_{n+1}=t_n+5$ $t_0=-3$

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

c) $t_{n+1}=3t_n$ $t_1=4$

The first order difference equation does not define an arithmetic sequence as the $t_n$ 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) $-22, -15, -8, -1, ...$

First we must check for a common difference.

$b=t_2-t_1$
$b=-15-(-22)$
$b=7$

$b=t_3-t_2$
$b=-8-(-15)$
$b=7$

$b=t_4-t_3$
$b=-1-(-8)$
$b=7$

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

$t_{n+1}=t_n+b$ $t_{n+1}=t_n+7$ $t_1=-22$

b) $23, 18, 13, 8, ...$

Again, we must check for a common difference.

$b=t_2-t_1$
$b=18-23$
$b=-5$

$b=t_3-t_2$
$b=13-18$
$b=-5$

$b=t_4-t_3$
$b=8-13$
$b=-5$

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

$t_{n+1}=t_n+b$ $t_{n+1}=t_n-5$ $t_1=23$

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