Growth & Decay with Sequences
Learn growth and decay with sequences for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). When a quantity changes by the same amount each period it forms an arithmetic sequence; when it changes by the same percentage it forms a geometric sequence.
You will learn to decide which model fits, use a common difference for steady change and a ratio such as 1.05 or 0.8 for percentage change, and work out future values for salaries, savings, depreciation and populations — skills that link straight into financial decisions.
Theory
Growth and decay problems use sequences: a fixed amount each period is arithmetic (linear), a fixed percentage is geometric (exponential). This Year 12 Mathematics Advanced topic (MAV-12-03) covers compound interest and depreciation.
A quantity that changes by the same amount each period forms an arithmetic sequence \(a_n=a+(n-1)d\) (a straight line). A quantity that changes by the same percentage forms a geometric sequence \(a_n=ar^{\,n-1}\) (a curve).
Compound growth: value after \(n\) periods \(=P(1+r)^{n}\). Depreciation: value \(=P(1-r)^{n}\).
Method
- Decide linear (fixed amount \(\to\) AP) or percentage (fixed rate \(\to\) GP).
- Use \(P(1+r)^n\) for growth, \(P(1-r)^n\) for depreciation.
- Substitute the number of periods \(n\).
| \(A\) | \(=\) | \(1000(1.05)^3\) |
| \(\approx\) | \(\$1157.63\) |
| \(A\) | \(=\) | \(20000(0.85)^2\) |
| \(=\) | \(\$14\,450\) |
| \(\text{A}\) | \(=\) | \(40000+2(3000)=46000\) |
| \(\text{B}\) | \(=\) | \(40000(1.05)^2=44100\) |
| \(\text{yr }1\) | \(=\) | \(5000(1.08)=\$5400\) |
| \(\text{yr }10\) | \(\approx\) | \(\$10\,794.62\) |
Common pitfalls
Frequently asked questions
What is the difference between linear and percentage growth?
Linear growth adds the same amount each period and forms an arithmetic sequence (a straight line). Percentage growth multiplies by the same factor each period and forms a geometric sequence (a curve).
What is the compound interest formula?
The value after n periods is P times (1 plus r) to the power n, where P is the principal and r is the interest rate per period as a decimal.
How do you calculate depreciation?
Multiply by (1 minus r) for each year. A 15 percent yearly loss means multiplying by 0.85 each year, so after 2 years the value is P times 0.85 squared.
Is compound interest arithmetic or geometric?
It is geometric, because the balance is multiplied by the same factor (1 plus r) each period.