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.

Finding the interest earned and the rule for the $n$th balance

For simple interest the total interest after $n$ periods is just $n$ lots of the fixed amount, $I = n\,d = n \times \frac{r}{100}\times P$, and the balance is the straight-line rule $V_n = P + n\,d$. For compound interest the interest earned is the balance minus the principal, $I = V_n - P = P(R^{\,n} - 1)$, because every dollar of interest is itself reinvested. This single difference, adding a constant versus multiplying by a constant, is the whole story of the topic, and it is why a graph of simple interest is a set of points on a line while compound interest curves upward.

A useful diagnostic in the exam is to ask which model the recurrence describes. A recurrence of the form $V_{n+1} = V_n + d$ (a constant added) is simple interest and arithmetic; a recurrence of the form $V_{n+1} = R\,V_n$ (a constant multiplier) is compound interest and geometric. If a question gives you a recurrence and asks for the type of growth, name it before you compute anything, because the marks usually follow the correct identification.

Converting between nominal and effective rates

A nominal annual rate quoted with a compounding frequency is not the rate your money actually grows at. To convert a nominal rate $r\%$ compounding $k$ times a year into the effective annual rate, compound one year's worth of periods and subtract one:

For example, a nominal $8\%$ compounding monthly has effective rate $\left(1 + \frac{0.08}{12}\right)^{12} - 1 = 8.30\%$. The effective rate always sits at or above the nominal rate, and the gap widens as compounding becomes more frequent. When a VCAA question asks you to compare two accounts, convert both to effective annual rates first; the higher effective rate is the better investment (or the worse loan).

Reading the question's period carefully

Most marks lost on this dot point come from a mismatch between the rate's period and the number of periods. If interest compounds quarterly for three years, that is $n = 3 \times 4 = 12$ periods at a rate of $\frac{r}{4}\%$ each. Always set the period rate and the period count together, then apply $V_n = P R^{\,n}$. The CAS finance solver does this for you when $P/Y$ and $C/Y$ are set, but in the technology-free Exam 1 you must do it by hand, so practise the arithmetic.

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.