Simple interest is calculated on the original principal only — the interest amount is the same each period.
$$I = PRT$$
$$A = P(1 + RT)$$
Where:
- $P$ = principal (initial amount)
- $R$ = interest rate per period (as a decimal)
- $T$ = number of periods
- $A$ = total amount (principal + interest)
$$V_{n+1} = V_n + \frac{r}{100} \times V_0, \quad V_0 = P$$
Or equivalently: $V_{n+1} = V_n + d$ where $d = \frac{r}{100} \times P$ (constant addition each period).
This is an arithmetic sequence with common difference $d$.
Example: \$3000 at 5% p.a. simple interest.
$d = 0.05 \times 3000 = 150$
$V_{n+1} = V_n + 150, \quad V_0 = 3000$
After 4 years: $V_4 = 3000 + 4(150) = 3600$
Compound interest is calculated on the current balance — interest earns interest.
$$A = P\left(1 + \frac{r}{100}\right)^n$$
$$V_{n+1} = \left(1 + \frac{r}{100}\right) \times V_n, \quad V_0 = P$$
This is a geometric sequence with ratio $R = 1 + \frac{r}{100}$.
Example: \$3000 at 5% p.a. compound interest.
$V_{n+1} = 1.05 \times V_n, \quad V_0 = 3000$
| Year | Balance |
|---|---|
| 0 | \$3000.00 |
| 1 | \$3150.00 |
| 2 | \$3307.50 |
| 3 | \$3472.88 |
| 4 | \$3646.52 |
Compare simple interest: \$3600 after 4 years. Compound yields more.
| Feature | Simple | Compound |
|---|---|---|
| Interest calculated on | Original principal | Current balance |
| Growth type | Linear (arithmetic) | Exponential (geometric) |
| Recurrence type | $V_{n+1} = V_n + d$ | $V_{n+1} = R \cdot V_n$ |
| Formula | $A = P(1+RT)$ | $A = P(1+r)^n$ |
| Better for investor | Short term | Long term |
Interest rates must match the compounding period:
| Quoted rate | Compounding | Rate per period |
|---|---|---|
| 12% p.a. | Annually | 12% per year |
| 12% p.a. | Monthly | 1% per month |
| 12% p.a. | Quarterly | 3% per quarter |
| 12% p.a. | Fortnightly | $\frac{12}{26}\%$ per fortnight |
KEY TAKEAWAY: Simple interest uses arithmetic sequences; compound interest uses geometric sequences. For any given rate and time, compound interest always gives more growth than simple interest (except at $n=1$).
EXAM TIP: Always convert the interest rate to match the compounding period. If compounding monthly, divide the annual rate by 12 and count months (not years) for $n$.
COMMON MISTAKE: Using the annual rate when compounding is monthly or quarterly. Always check: “What is the interest rate per compounding period?”
