Recursion in mathematics describes processes where each term is defined using the previous term(s). In VCE General Mathematics, recursion is applied extensively to financial modelling — understanding how money grows, shrinks, or changes over time.
Financial quantities change over time in a step-by-step fashion:
- Each year’s balance depends on last year’s balance
- Each repayment reduces the outstanding loan amount
- Each interest payment adds to the savings account
This “next value depends on current value” structure is perfectly captured by recurrence relations.
| Scenario | What happens each period |
|---|---|
| Simple interest | Fixed amount added each period |
| Compound interest | Balance multiplied by a constant each period |
| Reducing balance loan | Interest added, repayment subtracted |
| Annuity (investment) | Interest added, regular withdrawal made |
| Annuity (savings) | Regular deposit added, interest earned |
| Perpetuity | Interest earned equals withdrawal (balance stays constant) |
| Symbol | Meaning |
|---|---|
| $V_n$ | Value after $n$ periods |
| $V_0$ | Initial value (starting balance) |
| $r\%$ | Interest rate per period (as a percentage) |
| $R$ | Recurrence ratio $= 1 + \frac{r}{100}$ for compound; $= 1 - \frac{r}{100}$ for depreciation |
| $d$ | Regular payment amount (deposit or withdrawal) |
| $n$ | Number of periods |
In VCAA General Mathematics, technology (CAS calculator, financial calculator, or spreadsheet) is used to:
- Generate sequences term by term
- Find values after many periods
- Solve for unknown variables (interest rate, payment size, time to payoff)
KEY TAKEAWAY: Recursion turns complex financial problems into a simple rule: “apply the rule, get the next value, repeat.” Technology makes it practical to find values after dozens of periods.
VCAA FOCUS: You must be able to write the recurrence relation, interpret it in context, and use it to solve problems. Being able to identify the type of financial model from a recurrence relation is also assessed.
