Model simple interest with a linear recurrence and compound interest with a geometric recurrence, find balances and interest earned, convert between nominal and effective annual rates, and compare simple and compound growth

What this dot point is asking

VCAA wants you to model the two basic ways money grows. Simple interest adds a fixed amount each period and follows a linear (arithmetic) recurrence. Compound interest multiplies the balance each period and follows a geometric recurrence. You find balances and interest earned, handle compounding more often than yearly, and compare the two using an effective annual rate. This is the foundation for the loan and annuity work that follows.

Simple interest as a linear recurrence

With principal $P$ and an interest rate of $r\%$ per period, simple interest adds the constant amount $d = \frac{r}{100}\times P$ every period. The recurrence and rule are

The balance grows in a straight line, so a graph of balance against time is a set of points on a line.

Compound interest as a geometric recurrence

Compound interest multiplies the balance by the growth factor $R = 1 + \frac{r}{100}$ each period:

The balance grows geometrically, faster and faster, because each period's interest is added to the base for the next period.

Compounding more often than yearly

If interest compounds $k$ times a year at a nominal annual rate $r\%$, use the rate per period $\frac{r}{k}\%$ and count the number of periods. For example $12\%$ per annum compounding monthly uses $1\%$ per month over $12$ periods a year.

Why this matters for the exams

Interest questions open the financial modelling section and reward correctly identifying linear versus geometric growth and matching the rate to the compounding period. The recurrence form connects straight to the sequences dot point, and the geometric model becomes the engine of reducing-balance loans and annuities, where the finance solver takes over the arithmetic.