Financial Mathematics: Sequences, Series and Loans
Core ideas
Financial mathematics in Advanced builds on arithmetic and geometric sequences. Simple interest grows linearly; compound interest grows geometrically with factor (1 + r) per period, where r is the rate per period as a decimal. Future value and present value, depreciation, and annuities all reduce to geometric series.
For a recurring payment or investment, the balance after n periods satisfies a recurrence such as Aₙ₊₁ = (1 + r)Aₙ + M, where M is the payment (negative for a repayment). Solving these recurrences, or summing the resulting geometric series, gives closed forms. Reducing-balance loans are the standard application: the repayment is chosen so the balance reaches zero after the term.
Traps
- The rate and the number of periods must match: 6% per annum compounded monthly means r = 0.06/12 and n = 12 × years.
- Payments are usually at the end of each period; check whether the question implies a payment at the start.
- Interest earned is the final balance minus total contributions, not the final balance.
- Depreciation may be straight-line or reducing-balance - read which.
- Round only at the end; premature rounding shifts answers by dollars.
Worked example
$5000 is invested at 4% per annum compounding annually, with $800 added at the end of each year. After two years: 5000(1.04) = 5200, plus 800 gives 6000; then 6000(1.04) = 6240, plus 800 gives 7040. Equivalently A₂ = 5000(1.04)² + 800(1.04) + 800.
Rule of thumb
Draw a timeline for every finance question: mark each payment, convert the rate to a per-period value, and count periods between events carefully.