Recursion is a process where each step depends on the previous step. A recurrence relation is a rule that defines each term of a sequence from the previous term(s).
A first-order linear recurrence relation has the form:
$$V_{n+1} = R \cdot V_n + d, \quad V_0 = \text{starting value}$$
Where:
- $V_n$ = value at step $n$
- $R$ = constant multiplier (ratio)
- $d$ = constant added each step
- $V_0$ = initial value
| $R$ | $d$ | Type |
|---|---|---|
| 1 | positive $d$ | Arithmetic: increases by $d$ each step |
| 1 | negative $d$ | Arithmetic: decreases each step |
| $R > 1$ | 0 | Geometric: grows exponentially |
| \$0 < R < 1$ | 0 | Geometric: decays to zero |
| $R > 1$ | $d < 0$ | Compound growth with withdrawals |
| $R > 1$ | $d > 0$ | Compound growth with deposits |
$V_{n+1} = V_n + 5, \quad V_0 = 10$
Generates: 10, 15, 20, 25, 30, …
This is an arithmetic sequence with common difference 5.
$V_{n+1} = 1.06 \times V_n, \quad V_0 = 1000$
Generates: 1000, 1060, 1123.60, 1191.02, …
This is a geometric sequence with ratio 1.06 (6% growth each step).
$V_{n+1} = 1.05 \times V_n - 200, \quad V_0 = 5000$
| $n$ | $V_n$ |
|---|---|
| 0 | 5000.00 |
| 1 | \$1.05 \times 5000 - 200 = 5050.00$ |
| 2 | \$1.05 \times 5050 - 200 = 5102.50$ |
| 3 | \$1.05 \times 5102.50 - 200 = 5157.63$ |
Steps:
1. Identify $V_0$ (the starting value)
2. Identify what happens each period (interest? deposit? withdrawal?)
3. Express $V_{n+1}$ in terms of $V_n$
Example: \$2000 invested at 4% p.a. compound interest, \$100 deposited each year.
$V_{n+1} = 1.04 \times V_n + 100, \quad V_0 = 2000$
To find $V_5$, either:
- Apply the recurrence rule 5 times (by hand for small $n$)
- Use CAS: enter the recurrence relation and generate the sequence
KEY TAKEAWAY: Every first-order linear recurrence relation has the form $V_{n+1} = RV_n + d$. The two parameters $R$ and $d$ completely determine the behaviour of the sequence.
EXAM TIP: VCAA often gives a financial context and asks you to write the recurrence relation. Identify $R$ (the multiplier: compound factor) and $d$ (the additive part: deposit or repayment, with sign).
COMMON MISTAKE: Getting the sign of $d$ wrong. For a withdrawal or repayment, $d$ is negative. For a deposit or payment received, $d$ is positive.
