June 2026
The 2025 VCE General Mathematics exams showed that Recursion and Financial Modelling was not simply a matter of entering numbers into a CAS calculator.
Students needed to understand the model.
They needed to recognise whether a situation involved simple interest, compound interest, flat rate depreciation, reducing balance depreciation, recurrence relations, explicit rules, amortisation tables, effective interest rates or finance solver outputs. They also needed to manage signs, rounding, cash-flow direction and the exact form requested by the question.
This was especially clear in Examination 2, where students had to show working and explain financial processes.
Many errors were not caused by unfamiliar formulas.
They were caused by imprecise execution.
Was the balance reducing?
Was the interest calculated on the original amount or the current balance?
Was the question asking for a recurrence relation or a rule?
Should the answer be rounded?
Was the payment positive or negative in the finance solver?
Should the table values or solver values be used?
What does the final value actually represent?
These details determined the marks.
Simple interest required a simple model
Examination 1 Question 17 asked students to choose the expression for the balance after Dani invested $4000 for three years at 4% per annum simple interest.
The correct expression was:
4000 + 3(0.04 × 4000)
This is the defining feature of simple interest.
The interest is calculated on the original principal each year, not on an increasing balance.
A tempting incorrect response used 4000 × 1.04³, which would be appropriate for compound interest, not simple interest.
This question shows why students need to identify the financial model before calculating.
Simple interest and compound interest are not interchangeable.
The context tells students which structure to use.
Recurrence relations generated sequences
Examination 1 Question 18 asked about a recurrence relation of the form:
u₀ = a, uₙ₊₁ = Ruₙ + d
with a > 0, R = 0.5 and d = 0.
This becomes:
uₙ₊₁ = 0.5uₙ
Each term is half the previous term. Since the starting value is positive, the sequence decreases geometrically.
The correct description was geometric and decreasing.
This question rewarded students who could connect the recurrence relation to the behaviour of the sequence.
If R multiplies the previous term and d = 0, the sequence is geometric. If 0 < R < 1, the positive terms decrease.
Students should not treat recurrence relations as abstract notation only.
They describe a pattern of change.
Flat rate and reducing balance depreciation behaved differently
Examination 1 Question 19 compared the value of an asset using flat rate and reducing balance depreciation.
The table showed the same original value of $60 000, but different values after each year.
Under flat rate depreciation, the value decreased by the same dollar amount each year:
60 000, 56 000, 52 000, 48 000
Under reducing balance depreciation, the value decreased by a percentage of the current balance:
60 000, 55 200, 50 784, 46 721.28
Students needed to determine when the flat rate value first became lower than the reducing balance value.
This required understanding the two depreciation patterns.
Flat rate depreciation subtracts a constant amount.
Reducing balance depreciation subtracts a changing amount because it is calculated on the current reduced balance.
The comparison was not about which method always gives a lower value.
It depended on the year.
Reducing balance loans required current-balance thinking
Examination 2 Question 7 concerned Declan, a filmmaker and content creator, who had taken out a reducing balance loan for a new production.
The amortisation table showed:
- original balance: $850 000
- payment 1: $15 730.88
- interest on payment 1: $2975.00
- principal reduction for payment 1: $12 755.88
- balance after payment 1: $837 244.12
- interest on payment 2: $2930.35
Question 7b asked why the interest associated with payment 2 was lower than the interest associated with payment 1.
The answer was that the interest was calculated on a lower reducing balance.
This is the essential logic of an amortising loan.
As the balance decreases, the interest component of the payment decreases, even if the payment amount stays the same.
Students needed to explain the financial process, not just state that the number was lower.
Tables had to be used when instructed
Question 7c asked students to complete entries in the amortisation table.
The report noted that students were instructed to use the values in the table. Using a finance solver produced a different balance to the nearest cent and was not appropriate.
This is a very important lesson.
Sometimes CAS is not the correct tool, even when it is available.
If a question gives a table and asks students to use its values, the student must work from the table. Rounding in the table may affect the next values. A finance solver may calculate using full internal precision and produce a slightly different result.
For Question 7c, the required values were:
interest: 2885.55
principal reduction: 12 845.33
balance: 811 598.26
The point was not merely to find a mathematically possible balance.
It was to follow the table’s process.
Finance solver entries needed interpretation
Question 7d asked students to determine the payment number before the final payment of the loan.
The report gave finance solver entries:
N = 59.9999 ≈ 60
I% = 4.2
PV = 850 000
PMT = −15 730.88
FV = 0
P/Y & C/Y = 12
There were 60 payments in total, so there were 59 payments of $15 730.88 before the final payment.
This question required students to interpret the solver output.
The calculator produced approximately 60 total payments. The question asked for the payment number before the final payment, so students needed to subtract one.
A student who simply wrote 60 missed the wording.
Finance questions often require interpretation after the solver has done the calculation.
Recurrence relation and rule were not the same
Examination 2 Question 8 asked students to write a recurrence relation and a rule for a depreciation model.
The assessment guide gave:
V₀ = 40 000, Vₙ₊₁ = Vₙ − 8000
and:
Vₙ = 40 000 − 8000n
These are related, but they are not the same.
The recurrence relation defines each term from the previous term.
The rule gives the value directly for term n.
The report noted that many students did not understand the difference and gave a similar answer again.
This is one of the most important Recursion and Financial Modelling lessons from 2025.
When the question asks for a recurrence relation, students must include the starting value and a next-term relationship.
When the question asks for a rule, students must give an explicit expression in terms of n.
Model form matters.
Percentage depreciation had to be identified from change
Question 8b required students to identify a percentage depreciation rate. The assessment guide gave the answer as 20%.
This kind of question requires students to connect change in value to the original or current value, depending on the depreciation model.
Students should be able to recognise that a decrease of $8000 from $40 000 represents:
8000 ÷ 40 000 = 0.20 = 20%
Even if the model is written algebraically, the percentage interpretation needs to be understood.
The formula alone is not enough.
The number has to mean something.
Effective interest rates required correct substitution
Examination 2 Question 9 involved interest conversion.
The assessment guide gave a rate of 0.15% in Question 9a, then finance solver entries in Question 9b.i:
N = 52
I% = 6.960000
PV = 50 000
PMT = −75
FV = −49 565.34
P/Y & C/Y = 52
The effective annual rate was 6.96%.
Question 9b.ii required a recurrence multiplier:
R = 1.0013384…
rounded to 1.0013.
The assessment guide specified that students needed to show a calculation using the previous answer.
This matters because an effective rate question is not only about knowing the formula. Students need to use the correct rate, the correct number of compounding periods and the correct rounding instruction.
In financial modelling, small percentage errors compound quickly.
Rounding instructions differed by question
The assessment guide for the finance questions included specific rounding expectations.
For Question 9b.i, rounding to 2 decimal places applied.
For Question 9b.ii, rounding to 4 decimal places applied.
For Question 7c, the designated table values were required.
This reinforces a broader General Mathematics principle.
Students must read rounding instructions every time.
The same topic area may require dollars to the nearest cent, rates to two decimal places, multipliers to four decimal places, or no rounding at all.
Rounding is not a habit.
It is an instruction.
Finance solver signs affected answers
Examination 2 Question 10 was one of the most difficult finance questions.
The report showed finance solver entries involving:
N = 40
I% = 6.4
PV = −650 000
PMT = 22 126.27
FV = 0
P/Y & C/Y = 4
Then for a 20-period situation:
N = 20
I% = 6.4
PV = −650 000
PMT = 22 126.27
FV = 376 159.4283…
P/Y & C/Y = 4
The recurrence relation was:
D₀ = 376 159.43, Dₙ₊₁ = 1.016 × Dₙ − 22 126.27
The report noted that some students found the payment correctly from the calculator but answered with an incorrect positive value.
This is a classic finance-solver issue.
Money paid out and money received should have opposite signs. The sign convention matters because it reflects the direction of cash flow.
Students should not treat the calculator output as a number detached from the financial situation.
They need to know whether the value represents a debt, payment, present value or future value.
Quarterly interest had to become a recurrence multiplier
In Question 10, the annual interest rate was 6.4% with quarterly periods. Therefore, each quarter used:
6.4% ÷ 4 = 1.6%
The recurrence multiplier was:
1 + 0.016 = 1.016
That is why the recurrence relation used:
Dₙ₊₁ = 1.016 × Dₙ − 22 126.27
This shows how finance solver outputs and recurrence models connect.
The outstanding debt grows by quarterly interest, then decreases by the repayment.
Students need to understand the sequence:
Start with the debt.
Apply interest.
Subtract the payment.
Repeat.
This is recursion in a financial context.
The payment amount needed to be placed correctly
In recurrence relations for loans, students often make sign or placement errors.
For a debt model such as:
Dₙ₊₁ = 1.016 × Dₙ − 22 126.27
the debt first increases by interest, then decreases by the payment.
If students add the payment instead of subtracting it, the debt increases incorrectly. If they apply the payment before interest when the model assumes interest first, the balance changes. If they use the wrong compounding period, the multiplier is wrong.
The recurrence relation is not just algebra.
It describes a financial process.
Examination 2 required working for method marks
The assessment guides emphasised that when a question requires working, partial marks can be awarded for correct key steps.
This matters in finance and recursion because questions often involve multiple stages.
For example, a student may set up the finance solver correctly but misinterpret the final payment number. Or they may correctly find the effective rate but round incorrectly. Or they may write the correct multiplier but make an error in the initial value.
Showing working allows some of the reasoning to be credited.
A final answer alone is risky, especially in multi-step finance questions.
Students should show:
- finance solver entries
- rate conversions
- recurrence relation structure
- substituted values
- rounding step
- interpretation of the final output
This is not just presentation.
It protects marks.
CAS output needed to be converted into mathematical form
Students often rely heavily on CAS in this area, which is appropriate. But the 2025 reports show that students still needed to translate calculator output into the answer form required.
A finance solver might provide N, PMT or FV, but the question may ask for:
- number of payments before final payment
- a recurrence relation
- a rule
- a balance after a particular payment
- an effective annual interest rate
- an explanation of why interest changed
- a rounded value to a specified number of decimal places
The calculator provides data.
The student provides the answer.
Those are not the same thing.
Financial modelling questions were contextual
The 2025 finance questions involved realistic contexts: a filmmaker’s loan, depreciation of an asset, interest calculations and debt modelling.
This context mattered because it gave meaning to the numbers.
A balance is reducing because repayments reduce the principal.
Interest is lower because it is calculated on a lower balance.
A payment is negative or positive depending on cash-flow direction.
A depreciation model shows value declining over time.
A recurrence relation models a repeated process.
Students should always ask what the number represents.
Is it money owed?
Money paid?
Asset value?
Interest?
Principal reduction?
Remaining balance?
This prevents many errors.
Why Recursion and Financial Modelling errors happen
Errors in this area often happen because students use procedures without understanding the model.
They choose compound interest for a simple interest question.
They confuse recurrence relations with explicit rules.
They use a finance solver when the table values should be used.
They leave a solver output without interpreting the wording.
They use the wrong sign for a payment.
They round at the wrong time.
They write a recurrence relation that does not match the financial process.
They treat depreciation as the same under flat rate and reducing balance methods.
These mistakes are avoidable with model awareness.
Students should not begin by pressing buttons.
They should begin by identifying the model.
What future General Mathematics students should learn from 2025
The 2025 VCE General Mathematics exams show that Recursion and Financial Modelling preparation needs to focus on structure, interpretation and technology control.
Students should practise:
- distinguishing simple interest from compound interest
- recognising arithmetic and geometric sequences from recurrence relations
- comparing flat rate and reducing balance depreciation
- explaining why interest decreases in a reducing balance loan
- using table values when instructed
- interpreting finance solver outputs
- distinguishing total payments from payments before the final payment
- writing recurrence relations with starting values
- writing explicit rules in terms of n
- calculating percentage depreciation
- converting rates to effective rates
- writing recurrence multipliers correctly
- using correct finance-solver sign conventions
- applying rounding instructions exactly
- showing working in Examination 2
These skills make finance questions much more manageable.
The formulas matter, but the model matters more.
How ATAR STAR approaches Recursion and Financial Modelling
At ATAR STAR, Recursion and Financial Modelling is taught as model selection first.
Students learn to identify the financial process before using formulas or CAS. They practise simple and compound interest, depreciation, recurrence relations, annuities, loans, effective rates and finance solver interpretation with close attention to signs, rounding and context.
The 2025 Examination Reports confirm why this matters. High-scoring students did not simply use technology.
They understood the financial model behind the technology.