Practical Situations Represented by First Order Difference Equations

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 nth 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).


We now need to calculate the next term.

A_{n+1}=A_n+3% of 20,000

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.





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 nth 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).


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

See also

First Order Difference Equations
Generating the Terms of a Sequence Defined by a First Order Difference Equation
The Relationship between Difference Equations and Arithmetic Sequences
The Relationship between Difference Equations and Geometric Sequences