Some real life situations can be represented by first order difference equations.

A description of the situation (such as the pattern and starting term) are provided in these practical situations. You will have to interpret the information to understand what the question is asking and how to find the solution.

## Time Dependent Situations

Situations are time dependent when the starting term is ‘time zero’. This is represented by $t_0$ where $t_1$ is the term after the first period and $t_2$ is after the second time period.

### Example 1

Sally has invested \$20,000 with her bank and earns 3% simple interest per annum.

a) Describe this situation using a first order difference equation.

As we are creating an equation, we must define what terms we are using.

Let $A_n$ equal the amount Sally’s investment would be worth after the $n$th year.

We are told the initial investment. As this is a time dependent scenario, we use $A_0$ (not $t_0$ as we have defined this equation as $A_n$).

$A_0=20,000$

We now need to calculate the next term.

$A_{n+1}=A_n+3%$ of $20,000$
$A_{n+1}=A_n+600$

We know that this is an arithmetic sequence as there is a common difference (simple interest is calculated on the original amount so it will increase by the same amount). Now all we have to do is write the equation.

$A_{n+1}=A_n+600$   $A_0=20,000$

b) Calculate how much money Sally will have in her account after three years (assuming no withdrawals were made).

We can use the difference equation to calculate the first three terms.

$A_{n+1}=A_n+600$

$A_{1}=A_0+600$
$A_1=20,000+600$
$A_1=20,600$

$A_2=A_1+600$
$A_{2}=20,600+600$
$A_2=21,200$

$A_{3}=A_2+600$
$A_{3}=21,200+600$
$A_3=21,800$

After three years, Sally will have \$21,800 in her account.

## Ordinal Situations

Situations are ordinal where there is a position or rank in sequential order such as places or group numbers. This is represented by $t_1$ where $t_2$ is the next term in the sequence.

### Example 2

When analysing a netball tournament, the coach realised that Stephanie scored an extra 2 goals every game. In her first game Stephanie scored 1 goal.
What first order difference equation can be used to describe this scenario?

As we are creating an equation, we must define what terms we are using.

Let $G_n$ equal the amount of goals scored in the $n$th game.

We are told the amount of goals scored in the first game. As this is an ordinal scenario, we use $G_1$ (not $t_1$ as we have defined this equation as $G_n$).

$G_1=1$

We are told that Stephanie scores an additional 2 goals each game, so our equation needs to show that Stephanie’s next games goal total is the previous games score plus 2.

$G_{n+1}=G_n+2$   $G_1=1$