12 FURTHER MATHEMATICS Modelling linear growth and decay

Linear growth and decay is commonly found around the world. They occur when a quantity increases or decreases by the same amount at regular intervals. Everyday examples include the paying of simple interest or the depreciation of the value of a new car by a constant amount each year.

Modelling linear growth and decay A recurrence model for linear growth and decay The recurrence relations P o = 20, P n+1 = P n + 2 Q o = 20, Q n+1 = Q n − 2 both have rules that generate sequences with linear patterns, as can be seen from the table below. The first generates a sequence whose successive terms have a linear pattern of growth, and the second a linear pattern of decay.

Modelling linear growth and decay

As a general rule, if D is a constant, a recurrence relation rule of the form: V n+1 = V n + D can be used to model linear growth. V n − 1 = V n − D can be used to model linear decay. We are now in a position to use this knowledge to model and investigate simple interest loans and investments, as well as flat rate depreciation and unit cost depreciation of assets, the topics of this section.

Modelling linear growth and decay Simple interest loans and investments If you deposit money into a bank account, the bank is effectively borrowing money from you. The bank will pay you a fee for using your money and this fee is called interest. If a fixed amount of interest is paid into the account at regular time periods, it is called a simple interest investment. If you borrow money from the bank and are charged a fixed amount of interest after regular time periods, it is called a simple interest loan.

Modelling linear growth and decay Simple interest is a special case of linear growth in which the starting value is the amount borrowed or invested. The amount borrowed or invested is called the principal. The amount added at each step is the interest and is usually a percentage of this principal.

Modelling linear growth and decay

Once we have a recurrence relation, we can use it to determine things such as the value of the investment after a given number of years.

Modelling linear growth and decay

Depreciation Over time, the value of large item gradually decreases. A car bought new this year will not be worth the same amount of money in a few years’ time. A new television bought for $2000 today is unlikely to be worth anywhere near this amount in 5 years.

Modelling linear growth and decay Large equipment, machinery and other assets used in a business also lose value, or depreciate, over time. The depreciating value of equipment is often taken into account when calculating the actual costs of the business operations. It is important for businesses to be able to estimate the likely value of an asset after a certain amount of time. This is called the future value of the asset. After a certain amount of time, or when the value of an item is depreciated to a certain amount, called its scrap value, the item will be sold or disposed of. At this point, the item has reached the end of its useful life and will be written off. This means the item is no longer an asset for the business.

Modelling linear growth and decay Individuals often use the depreciation in value of equipment to calculate tax refunds. The taxation law allows individuals to buy equipment such as computers or other items necessary for their work and then to claim tax refunds based on the depreciating value of those assets. There are a number of techniques for estimating the future value of an asset. Two of them, flat-rate depreciation and unit-cost depreciation, can be modelled using a linear decay recurrence relation.

Modelling linear growth and decay Flat-rate depreciation Flat-rate depreciation is very similar to simple interest, but instead of adding a constant amount of interest, a constant amount is subtracted to decay the value of the asset after every time period. This constant amount is called the depreciation amount and, like simple interest, it is often given as a percentage of the initial purchase price of the asset.

Modelling linear growth and decay

Unit-cost depreciation Some items lose value because of how often they are used, rather than because of their age. A photocopier that is 2 years old but has never been used could still be considered to be in ‘brand new’ condition and therefore worth the same, or close to, what it was 2 years ago. But if that photocopier was 2 years old and had printed many thousands of papers over those 2 years, it would be worth much less than its original value.

Modelling linear growth and decay Cars can also depreciate according to their use rather than time. People often look at the number of kilometres a car has travelled before they consider buying it. An older car that has travelled few kilometres overall could be considered a better buy than a new car that has travelled a large distance. When the future value of an item is based upon use rather than age, we use a unit-cost depreciation method. Unit-cost depreciation can be modelled using a linear decay recurrence relation.

Modelling linear growth and decay

WORK TO BE COMPLETED Exercise 8C – All Questions