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:
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.
