Recursion and financial modelling

Area of study: Recursion and financial modelling (VCE General Mathematics Units 3–4; formerly VCE Further Mathematics). Corpus: corpus/text\*.txt — every VCAA Further Mathematics / General Mathematics exam paper and examination report 2006–2026 (including NHT series), converted to plain text.

Every claim below is traceable to a named corpus file. Quotations preserve VCAA wording; the plain-text conversion replaces some symbols (× → � or x, ≈ → ` or≈`, subscripts flattened), and these are silently normalised in quotations but the words are verbatim.

Corpus gaps to be aware of
- Documents_exams_mathematics_2024_2024GeneralMaths1-w.txt and ...2024GeneralMaths2-w.txt are 0 bytes — the 2024 main-series exam papers are missing. 2024 question wording must be reconstructed from 2025-04_2024generalmathematics2-assessment-guide.txt, 2025-03_2024generalmaths2-report.txt, 2025-03_2024generalmaths1-report.txt and the assessor PL transcript 2025-04_General_Maths_Examination_2.txt.
- 2025-05_2025-NHT-generalmaths1.txt and 2025-05_2025-NHT-generalmaths2.txt are 0 bytes; use 2025-06_2025NHT-GeneralMath2-assessment-guide.txt and 2025-10_2025-NHT-general-maths1-report.txt / ...2-report.txt.
- There is no 2023 General Mathematics Exam 2 examination report in the corpus (only 2025-07_2023generalmaths1-report.txt for Exam 1).

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0. Exam architecture by era (essential context for the taxonomy)

0.1 2006–2015 — LEGACY (pre-compulsory)

Recursion/financial content was not in the Core. It sat in two optional Section B modules:

Legacy notation uses difference equation where the modern papers use recurrence relation, and writes the initial value second (Cn+1 = ... , C1 = 2000) or with a 1-based index. Legacy papers routinely used the 1 + r/100-style formula sheet, including effective rate of interest ≈ 2n/(n+1) × flat rate (on every formula sheet 2006–2015, e.g. Documents_exams_mathematics_2013_2013furmath2-w.txt line 1150).

0.2 2016–2019, 2021–2022 — Further Mathematics, compulsory Core

• Exam 1: Section A Core; Questions 17–24 = Recursion and financial modelling; 8 multiple-choice, 5 options (A–E), 1 mark each.
• Exam 2: Section A Core — Data analysis (24 marks) + Recursion and financial modelling (12 marks), typically 3 questions of 3–5 marks. Confirmed: "a compulsory Core section of Recursion and financial modelling (worth 12 marks)" — Documents_exams_mathematics_2016_FM2_examrep16.txt line 10; identical in 2017, 2018, 2019, 2021, 2022 reports.

0.3 2020 only — COVID-adjusted study design

• Exam 1: "the Core section comprised two components: Data analysis (Questions 1–20) and Recursion and financial modelling (Questions 21–30)" — Documents_exams_mathematics_2020_2020furmaths1-exam-report.txt line 10. 10 MCQ.
• Exam 2: "the compulsory Core section of Recursion and financial modelling (worth 15 marks)" — Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt line 4. 5 questions (Q7–Q11). One module only.

0.4 2023 onwards — General Mathematics (modules abolished)

From 2025-04_genmath-specs-w.txt lines 20–28:

"The examination will be divided into four content areas: data analysis, recursion and financial modelling, matrices, and networks and decision mathematics."
Exam 1: "40 multiple-choice questions worth 1 mark each. Of these 40 questions, 16 will be allocated to data analysis, 8 will be allocated to recursion and financial modelling, 8 … matrices, and 8 … networks."
Exam 2: "short-answer and extended-answer questions, including multi-stage questions. 24 marks … data analysis, 12 marks … recursion and financial modelling, 12 marks … matrices, 12 marks … networks."

• Exam 1 Q17–24 in every year 2023–2026.
• Option count changed: 2023 GM1 still had five options A–E (2025-07_2023generalmaths1-report.txt tables show % A … % E). From 2024 onwards Exam 1 has four options A–D (2025-03_2024generalmaths1-report.txt tables show % A | % B | % C | % D | % N/A; confirmed again 2025 and 2026 NHT papers).
• Exam 2 recursion = 12 marks, split across 3–4 questions.

0.5 Formula sheet (2016 onwards) — only two entries for this area

From 2025-11_2025-GeneralMaths1_0.txt lines 889–896 (identical 2016–2026):

Recursion and financial modelling

first-order linear recurrence relation      u0 = a,  un+1 = Run + d

effective rate of interest for a            r_effective = [ (1 + r/(100n))^n − 1 ] × 100%
compound interest loan or investment

There is no compound interest formula, no annuity formula and no depreciation formula on the sheet. Examiners repeatedly punish students who try to reconstruct one (see §3.7).

0.6 The rounding instruction changed in 2023 — critical

• 2016–2022 Exam 2 front matter (Instructions for Section A):

  "In Recursion and financial modelling', all answers should be rounded to the nearest cent unless otherwise instructed." (Documents_exams_mathematics_2016_2016furmath2-w.txt` line 68; identical wording at line 68 of 2017, 2018, 2019 papers and line 67 of 2020, 2021, 2022 papers.)

• 2023 onwards there is NO topic-specific rounding default. The global instruction is:

  "In all questions where a numerical answer is required, you should only round your answer when instructed to do so." (Documents_exams_mathematics_2023_2023genmath2-w.txt line 58; 2025-11_2025-GeneralMaths2.txt lines 43–45.)

This is the single biggest era difference for answer-form marking.

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1. Complete question-type catalogue

Index of Exam 2 recursion questions by year

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1.1 Sequences and recurrence relations

1.1a Generate terms from a given recurrence relation

Looks like: a bare recurrence relation and a request for a specific term, or the first four terms, or the term index at which a value appears. Marks: 1 (Exam 1 only, almost exclusively). Exam 1: very high. Exam 2: rare as a standalone — absorbed into "showing recursive calculations".

Examples: - 2016 Exam 1 Question 17 — "Consider the recurrence relation below. A0 = 2, An+1 = 3An + 1. The first four terms of this recurrence relation are …" (options 2, 7, 22, 67 ...). - 2019 Exam 1 Question 17 — "Consider the recurrence relation shown below. A0 = 3, An+1 = 2An + 4. The value of A3 in the sequence generated by this recurrence relation is given by …" with options left unevaluated (A. 2 × 3 + 4 … E. 2 × 52 + 4). Reused verbatim as GM Sample Exam 1 Q17. - 2020 Exam 1 Question 21 — "The following recurrence relation can generate a sequence of numbers. T0 = 10, Tn+1 = Tn + 3. The number 13 appears in this sequence as A. T1 … E. T13". (Answer is the index, not the value.) - 2022 Exam 1 Question 17 — "R0 = 2, Rn+1 = 2 − Rn. The value of R2 is …". Report: "R1 = 2 – 2 = 0; R2 = 2 – 0 = 2" (Documents_exams_mathematics_2022_2022furmaths1-report.txt). Only 44% correct. - 2023 Exam 1 Question 17 — "T0 = 5, Tn+1 = −Tn. The value of T2 is …" — only 43% correct (2025-07_2023generalmaths1-report.txt). - NHT 2023 Exam 1 Q17 — "T0 = 11, Tn+1 = Tn − 4. The first negative term in this sequence is A. T2 … E. T11".

Reverse variants (harder, later in the set): - 2021 Exam 1 Q17 — "L0 = 37, Ln+1 = Ln + C. The value of L2 is 25. The value of C is …" (answer −6). - NHT 2026 Exam 1 Q17 — "T0 = a, Tn+1 = 2Tn + 1. Given that T2 = 15, what is the value of a?" - NHT 2024 Exam 1 Q18 — "T0 = 5, Tn+1 = k − Tn. All terms of the sequence have the same value. The constant k is equal to …".

1.1b Write a recurrence relation from a worded description

Looks like: a financial scenario in prose, then the instruction template of §2.1. Marks: 1 in Exam 2 (2 marks in 2019 Q9c and 2022 NHT Q7c); 1 in Exam 1 as MCQ. Exam 2: appears in virtually every year. Exam 1 version is "A recurrence relation that can be used to determine … is".

Exam 2 examples: - 2016 Exam 2 Q6b (1 mark) — "Let Cn be the value of the caravan n years after it was purchased. Assume that the value of the caravan has been depreciated using the flat rate method of depreciation. Write down a recurrence relation, in terms of Cn+1 and Cn, that models the value of the caravan." Answer C0 = 38 000, Cn+1 = Cn − 2750. 33% correct. - 2019 Exam 2 Q9c (2 marks) — "Let Bn be the balance of the loan n months after these changes apply. Write down a recurrence relation, in terms of B0, Bn+1 and Bn, that could be used to model the balance of the loan over these four years." Answer B0 = 262 332.33, Bn+1 = 1.004Bn − 3517.28. 24% full marks. - 2022 Exam 2 Q7c (1 mark) — "Let Pn be the balance, in dollars, of Pina's annuity after n months. Write a recurrence relation, in terms of P0, Pn+1 and Pn, that can model this balance from month to month." Answer P0 = 540 000, Pn+1 = 1.0025 × Pn − 5214.28. 35% correct. - 2024 Exam 2 Q5c (1 mark) — Answer V0 = 15 000, Vn+1 = Vn − 60. 59% correct. - 2025 Exam 2 Q8a.i (1 mark) — "The value of the lighting equipment could be modelled by either a recurrence relation or a rule. i. Write a recurrence relation in terms of V0, Vn + 1 and Vn." Answer V0 = 40 000, Vn+1 = Vn − 8000; guide notes "BOTH parts must be correct". 73% correct.

Exam 1 MCQ examples: - 2016 Exam 1 Q19 — "The purchase price of a car was $26000. Using the reducing balance method, the value of the car is depreciated by 8% each year. A recurrence relation that can be used to determine the value of the car after n years, Cn, is …" (C0 = 26000, Cn+1 = 0.92 Cn). - 2019 Exam 1 Q19 — "Geoff purchased a computer for $4500. He will depreciate the value of his computer by a flat rate of 10% of the purchase price per annum." (V0 = 4500, Vn+1 = Vn − 450). Reused as Sample Exam 1 Q18. - NHT 2023 Exam 1 Q20 — "The value of a computer, purchased for $6000, is depreciated by a flat rate of 8% per annum. Which one of the following recurrence relations models the year-to-year value, Vn, of the computer?"

1.1c Identify the rule/recurrence from a sequence, table or graph

Marks: 1. Exam 1 dominant. - 2017 Exam 1 Q18 — "The first five terms of a sequence are 2, 6, 22, 86, 342 ... The recurrence relation that generates this sequence could be …" (P0 = 2, Pn+1 = 4Pn − 2). - 2017 Exam 1 Q22 — graph with labelled points (0, 7000) (1, 6580) (2, 6185.20) (3, 5814.09) (4, 5465.24); "This graph could show the value of A. a piano depreciating at a flat rate of 6% per annum … E. an annuity investment with additional payments of 6% …". - 2019 Exam 1 Q21 — compound-interest graph with a missing coordinate (2, b); "The value of b is". - 2020 Exam 1 Q25 / Sample Exam 1 Q20 — "The graph below represents the value of an annuity investment, An … A recurrence relation that could match this graphical representation is" (five relations differing in R and d). - 2018 Exam 1 Q20 — "The graph below shows the value, Vn, of an asset as it depreciates over a period of five months. Which one of the following depreciation situations does this graph best represent?" Report: "Students needed to recognise that the rate of reduction in value is constant in both phases of depreciation, indicating flat-rate depreciation, not reducing balance depreciation." - NHT 2021 Exam 1 Q17 — unlabelled value-vs-years graph; "The graph above could represent the value of A. the balance of a savings account earning simple interest … E. a reducing balance loan with interest-only repayments."

1.1d Classify what a recurrence relation models / structural properties

• 2016 Exam 1 Q20 — "V0 = 10000, Vn+1 = 1.04Vn + 500. This recurrence relation could be used to model A. a reducing balance depreciation … D. an annuity investment with periodic additions of $500 made to the investment. E. an interest-only loan of $10000."
• 2018 Exam 1 Q21 — "Which one of the following recurrence relations could be used to model the value of a perpetuity investment, Pn, after n months?" (test: R·P0 − d = P0).
• 2020 Exam 1 Q23 — four relations listed; "How many of these recurrence relations indicate that the value of an asset is depreciating?"
• 2022 NHT Exam 1 Q19 — "Which one of the following recurrence relations generates a sequence in which values grow geometrically?"
• NHT 2024 Exam 1 Q22 — "P0 = a, Pn+1 = RPn − d. All constants, a, R and d, are greater than 1. Four options of what the value of Pn could represent are listed below: a reducing balance loan / an annuity / an asset depreciated using the unit cost method / a perpetuity. How many of these four options could be represented by the recurrence relation?"
• 2025 Exam 1 Q18 — "A recurrence relation is of the form u0 = a, un+1 = Run + d. If a > 0, R = 0.5 and d = 0, the sequence generated will be A. arithmetic and increasing … D. geometric and decreasing." Report: "When d = 0 a geometric sequence exists. Reject options A and B. When 0 < R < 1 the sequence generated will be decreasing." 52% correct.
• 2024 Exam 1 Q17 — report comment: "For a geometric sequence, [d] always equals zero." 36% correct (lowest-scoring recursion MCQ of 2024).
• 2023 Exam 1 Q24 — "The following recurrence relation models the value, Pn, of a perpetuity after n time periods. P0 = a, Pn+1 = RPn − d. The value of R can be found by calculating …". Report: "A perpetuity has a constant value." Only 27% correct; 37% chose D.
• NHT 2023 Exam 1 Q21 — "The rule for the future value of an asset after n depreciation periods is Vn = 4000 − 20n. The method used for depreciating the value of the asset could be … E. either flat rate or unit cost."

1.1e Convert between recurrence relation and rule (both directions)

Marks: 1. Appears in both exams; this is a recurring high-failure item. - 2020 Exam 1 Q22 — "An asset is purchased for $2480. The value of this asset after n time periods, Vn, can be determined using the rule Vn = 2480 + 45n. A recurrence relation that also models the value of this asset after n time periods is …" (distractors use Vn = 2480 as the initial condition, or keep the n). - 2022 Exam 1 Q22 — "Tim deposited $6000 … A rule for the balance, Tn, in dollars, after n years is given by Tn = 6000 × 1.003^(12n). Let Rn be a new recurrence relation that models the balance of Tim's account after n months." - 2023 Exam 1 Q18 — "G0 = 15000, Gn+1 = Gn − 1314. A rule for Gn, the value of the machine after n years is …" (Gn = 15000 − 1314n). - NHT 2024 Exam 1 Q17 — "M0 = 60000, Mn+1 = 0.85 Mn. An equivalent rule to determine the value of the car after n years is …" (Mn = 60000 × 0.85^n). - 2022 Exam 2 Q6c (1 mark) — rule → recurrence: "The value of the equipment, in dollars, after n years, Vn, can also be modelled by a recurrence relation. Write this recurrence relation in terms of V0, Vn+1 and Vn." - 2025 Exam 2 Q8a (2 × 1 mark) — both directions in one question (recurrence in a.i, rule in a.ii). - NHT 2026 Exam 2 Q6b/6c — recurrence in years (b) and rule in kilometres (c) for the same asset.

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1.2 Simple interest / flat rate depreciation (linear growth and decay)

Three sub-forms are examined: recurrence form (V0 = a, Vn+1 = Vn ± d), rule form (Vn = a ± dn), and time-to-reach-a-value.

Recurrence and rule

• 2017 Exam 2 Q5 (5 marks) — "The value of Alex's van is depreciated using the flat rate method of depreciation. … V0 = 75000, Vn+1 = Vn − 3375. a. Recursion can be used to calculate the value of the van after two years. Complete the calculations below by writing the appropriate numbers in the boxes provided. (2 marks)"; then "b. i. By how many dollars is the value of the van depreciated each year? ii. Calculate the annual flat rate of depreciation in the value of the van. Write your answer as a percentage."
• 2020 Exam 2 Q7 (4 marks) — V0 = 120000, Vn+1 = Vn − 15000: "a. By what amount, in dollars, does the value of the machine decrease each year? b. Showing recursive calculations, determine the value of the machine, in dollars, after two years. c. What annual flat rate percentage of depreciation is used by Samuel? d. The value of the machine … could also be determined using a rule of the form Vn = a + bn. Write down this rule for Vn." (Q7d only 44% correct — "Some students were unsure of the distinction between a rule and a recurrence relation.")
• 2022 Exam 2 Q6 (4 marks) — starts from the rule: "Vn = 200000 − 12500n. a. Determine V1 … b. After how many years will the equipment first have a value of zero? c. … write this recurrence relation … d. … What name is given to this type of depreciation?"
• 2025 Exam 2 Q8 (3 marks) — flat-rate depreciation given only as a graph; write recurrence, write rule, give the annual flat rate percentage (20%).
• 2024 Exam 2 Q5 (4 marks) — rule in weeks, questions about per-week depreciation, value after 4 years, recurrence relation, and annual flat rate (20.8%).

Simple interest

• 2025 Exam 1 Q17 — "Dani invests $4000 for three years. The account earns simple interest at 4% per annum. The balance in the account after three years can be calculated using A. 4000 × 1.04³ … D. 4000 + 3(0.04 × 4000)." 53% correct.
• 2022 Exam 1 Q23 — compares a compound investment with a simple-interest recurrence: "The two investments will finish at the same value, rounded to the nearest cent, if Joseph's investment is modelled by which one of the following recurrence relations?" (J0 = 3500, Jn+1 = Jn + 267.72).
• 2024 Exam 1 Q20 — report: "Step 1. Calculate the interest for the compounding investment … Step 2. Calculate the simple interest rate." 48% correct.
• NHT 2022 Exam 1 Q17 — "Steve invested $4000 in a savings account that pays 2% per annum simple interest. After how many years will the savings account have earned a total of $400 in interest?"
• NHT 2026 Exam 1 Q23 — two simple-interest investors; "The amount by which Teresa's balance exceeds Mel's balance after n years, Dn, is given by … C. Dn = 190n D. Dn = 190n − 100."

Time to reach a value

• 2016 Exam 2 Q6a "Show that the average depreciation in the value of the caravan per year was $2750."
• NHT 2017 Exam 2 Q5c — "After how many years will the value of the snooker table first fall below $2000?" (answer 6 years).
• 2019 Exam 2 Q7c — "Phil plans to replace these tools when their value first falls below $20000. After how many years will Phil replace these tools?" (11 years, 79% correct).
• 2018 Exam 2 Q4b.ii — "After how many months will the balance of Julie's account first exceed $12300" (4 months).
• 2025 Exam 1 Q19 — a table comparing flat-rate and reducing-balance values; "After how many years will the value using flat rate depreciation first be lower than the value using reducing balance depreciation?" Report gives two methods, including "Solve 60 000 − 4000n = 60 000 × 0.92^n for n which gives n = 5.582…". 49% correct.

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1.3 Compound interest / reducing balance depreciation (geometric growth and decay)

Recurrence ↔ rate conversions (the highest-frequency 1-mark item in the whole topic)

• 2016 Exam 2 Q5c — "What is the annual percentage compound interest rate for this savings account?" from Vn+1 = 1.04Vn. Answer 4%. "Common incorrect answers were 1.04 or 0.04%." 65% correct.
• 2017 Exam 2 Q5c — "R0 = 75000, Rn+1 = 0.943Rn. At what annual percentage rate is the value of the van depreciated each year?" Answer 5.7%; "Some students gave 9.43% or 94.3%."
• 2018 Exam 2 Q5a — "For each of the first three years of reducing balance depreciation, the value of R is 0.85. What is the annual rate of depreciation in the value of the car during these three years?" Answer 15%. "Some students gave the R value of 0.85." Only 55%.
• 2019 Exam 2 Q7b — "What is the annual percentage rate of depreciation used by Phil?" (Vn+1 = 0.9Vn → 10%). "Incorrect answers included 0.9% and 90%."
• 2020 Exam 2 Q9b / Q10b — monthly rate vs annual rate: 9b answer 0.3% ("Many gave the annual interest rate percentage"); 10b answer 2.88% ("0.24% per month was sometimes given"). Mirror-image errors in the same paper.
• 2022 Exam 2 Q8b — "Determine the annual compound interest rate for this loan." (1.002 → 2.4%). "0.2% was a common incorrect answer."
• 2024 Exam 2 Q7c — (1.003 − 1) × 12 × 100 → 3.6%; "Those who were incorrect often gave 0.3%, as this was the monthly rate."
• 2018 Exam 1 Q18 / 2024 Exam 1 Q18 — report: "Interest rate = (1.00075 − 1) × 52 × 100 = 3.9%" (weekly).
• NHT 2021 Exam 1 Q18 — "V0 = 8000, Vn+1 = RVn. This investment earns compound interest at the rate of 5% per annum, compounding quarterly. What is the value of R?" (1.0125).
• NHT 2024 Exam 2 Q8c — "If the depreciation rate per month was 1.5%, what would be the value of R in this recurrence relation?" (0.985).

Rule form for compound growth

• 2016 Exam 2 Q5d.i — "The amount of money in the account after n years, Vn, can also be determined using a rule. Complete the rule below by writing the appropriate numbers in the boxes provided. Vn = □ × □^n". Answer Vn = 1.04^n × 15 000. Only 61%.
• 2018 Exam 2 Q4c.i — "A rule of the form Vn = a × b^n can be used … Complete this rule … balance = □ × □^n". "A common error was for students to write the two numbers in the incorrect boxes."
• 2020 Exam 1 Q26 / Sample Q21 — "Ray deposited $5000 … 3% per annum, compounding quarterly. A rule for the balance, Rn, in dollars, after n years is given by …" (Rn = 5000 × 1.0075^(4n)). Report: "The growth factor per quarter was … As interest was compounding quarterly, the number of compounding periods in n years is equal to 4n, which is the required power." One of the hardest 2020 items.
• NHT 2024 Exam 1 Q19 — "Edo invests $10000 in an account earning 3% interest per annum compounding monthly. The value, Vn, of Edo's investment after n months is given by …" (Vn = 10000 × 1.0025^n).
• NHT 2017 Exam 2 Q6c (2 marks) — "The value of Qn can be determined from a rule. Complete this rule by writing the missing values in the boxes provided below. Qn = □ × □^n" (answer 2080.05 × 1.0046^n).

Unit cost depreciation

Always paired with a "per unit" rate and a usage rate. - 2016 Exam 2 Q6c — "The caravan has travelled an average of 5000 km in each of the eight years … By how much is the value of the caravan reduced per kilometre travelled?" Answer $0.55/km. "A very common incorrect answer of $4.40 was found by ignoring the eight years over which the depreciation occurred." Only 29%. - 2017 Exam 1 Q21 — printer $680 → $125 after four years, 1920 pages/year. "The depreciation in the value of the printer, per page printed, is closest to …". Report: "Students who divided by 1920 only would get the result 0.289, which may have been incorrectly interpreted as 3 cents." - 2019 Exam 1 Q22 — "A rule for the value of the machine after n units are produced, Vn, is …" (Vn = 30000 − 0.16n). - 2021 Exam 2 Q7 (3 marks) — M0 = 12000, Mn+1 = Mn − 1440 with "$0.05 per cup": a. cups per year (28 800); b. equivalent annual flat rate (12%); c. "Complete the rule below that gives the value of the coffee machine, Mn, in dollars, after n cups have been produced. Mn = □ + □ × n". Q7c only 21% — "Many students did not recognise that the value was of the machine in terms of the number of cups of coffee made." - 2022 Exam 1 Q20 — "E0 = 100000, En+1 = En − 5475" with "used for 10 hours per day for all 365 days of the year": "The value of the equipment is depreciated by … A. $1.50 per hour." - 2023 Exam 1 Q19 — "The number of cups made by the machine per year is" (1314 ÷ 0.04 = 32 850). - 2025 Exam 1 Q20 — "Steve uses the gardening equipment for 960 hours per year. The value … is $7680 after two years. By what amount does the gardening equipment depreciate per hour used?" Report: "Loss of value = $12 000 − $7680 = $4320; Hours used = 960 × 2 = 1920; $4320 ÷ 1920 = $2.25." 55% correct. - NHT 2021 Exam 2 Q6 (3 marks) — V0 = 120000, Vn+1 = Vn − 10920 at "$3.50 per hour of use": "a. Use recursion to show that the value of the machinery after two years is $98 160. b. The machinery is used all 52 weeks of the year and for the same number of hours each week. For how many hours each week is the machinery used? c. … What annual percentage flat rate of depreciation is represented?" - NHT 2026 Exam 2 Q6 — "The value of the car is depreciated by 35 cents for each kilometre travelled. Assume he travels 25000 km each year." Recurrence in years (V0 = 60 000, Vn+1 = Vn − 8750), rule in kilometres (Wn = 60 000 − 0.35n).

Mixed-method comparisons

• 2020 Exam 1 Q29 — "The value of a van purchased for $45000 is depreciated by k% per annum using the reducing balance method. After three years … it is then depreciated in the fourth year under the unit cost method at the rate of 15 cents per kilometre. … The value of k is".
• NHT 2024 Exam 1 Q21 — Option 1 = 2 years flat rate then 2 years reducing balance; Option 2 = constant reducing balance rate for four years; "For both options to predict the same value after four years, the rate per annum used for Option 2 is closest to".
• 2023 Exam 1 Q20 & Q21 (paired stem: $3000 computer → $600 after four years) — "If Audrey uses flat rate depreciation, the depreciation rate, per annum is" (20%, 82% correct) / "If Audrey uses reducing balance depreciation, the depreciation rate, per annum is closest to" (33%, 48% correct).
• NHT 2026 Exam 1 Q20 — "She uses the reducing balance method and finds that the value has exactly halved after five years. The annual reducing balance depreciation rate used by Vinita is closest to" (12.9%).

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1.4 Compound interest investments with regular additions (annuity investment / savings plan)

Signature: V0 = a, Vn+1 = R·Vn + d with d > 0, described as "immediately after the interest is added/calculated, … deposits a further $X".

• 2016 Exam 1 Q23 — "Sarah invests $5000 … 3.9% per annum compounding quarterly. At the end of each quarter, immediately after the interest has been paid, she adds $200 to her investment. After two years, the value of her investment will be closest to".
• 2018 Exam 1 Q17 & Q18 (paired) — V0 = 46000, Vn+1 = 1.0034Vn + 500: "What is the value of the regular payment added to the principal of this annuity investment?" and "Between the second and third years, the increase in the value of this investment is closest to".
• 2018 Exam 1 Q24 — retirement goal $600 000; interest rate falls mid-way; "she must increase her monthly payment if she is to reach her retirement goal. The value of this new monthly payment will be closest to".
• 2021 Exam 1 Q22 — "She will make an extra payment into the account each month, immediately after the interest is calculated. The minimum value of this payment is closest to".
• 2020 Exam 2 Q9c (1 mark) — "If Samuel had deposited an additional $50 at the end of each month immediately after the interest was added, how much extra money would be in the savings account after one year? Round your answer to the nearest dollar." Answer $610. "Some students answered $600 because they did not consider the interest earned by these additional payments, or gave the new balance of $5793." 34% correct.
• 2022 Exam 1 Q24 — D1 = C, Dn+1 = Dn describing a constant quarterly addition; find C.
• 2025 Exam 1 Q22 — "After the interest is added each week, Rita deposits an additional $50 into the account. Assume there are exactly 52 weeks in one year. The annual interest rate, compounding weekly, that is required to achieve a balance of $14000 after three years is closest to". 68% correct.
• 2017 Exam 2 Q7b (2 marks) — perpetuity converted to annuity investment; "For the first four years Alex makes a further payment each month of $500 … This monthly payment is made immediately after the interest is added. After four years … he increases the monthly payment. This new monthly payment gives Alex a balance of $500000 … after a further two years. What is the value of Alex's new monthly payment? Round your answer to the nearest cent." Only 26% got full marks.
• NHT 2017 Exam 2 Q7 (3 marks) — "Immediately after the interest has been added each month, the community centre deposits a further $100 … a. What is the annual interest rate, compounding monthly, that is required to achieve this goal? Write your answer correct to two decimal places. b. … After 36 deposits, the community centre stopped making the additional monthly deposits."
• NHT 2019 Exam 2 Q7b/7c — write the recurrence for a $150/month addition; then "What annual interest rate would Tisha require? Round your answer to two decimal places."
• Legacy: 2013 Exam 2 Module 4 Q2b — "Suppose instead that at the end of each month Hugo added $200 to his initial investment of $5000. Find the value of this investment immediately after the 12th monthly payment of $200 is made."
• Legacy: 2015 Exam 2 Module 4 Q4b/4c — "account balance = 2000 × (1 + 0.008) + 200. b. Show that the annual compounding rate of interest is 3.2%. c. Determine the amount in Michael's account, after the $200 has been added, at the end of five years. Write your answer correct to the nearest cent."

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1.5 Reducing balance loans

Balance after n payments

• 2016 Exam 2 Q7a.i — "Find the amount that Ken will owe on his loan after he has made 12 repayments." Report gives the full solver block (N=12, I%=6.9, PV=70 000, PMT=−800, FV=−65 076.219…, P/Y=C/Y=12) and notes "Rather than use a financial solver … a number of students adopted a formulaic approach, almost always unsuccessfully, based on the compound interest formula."
• 2017 Exam 1 Q17 — "The value of a reducing balance loan, in dollars, after n months, Vn … V0 = 26 000, Vn+1 = 1.003Vn − 400. What is the value of this loan after five months?"
• 2022 Exam 1 Q18 — "The balance of the loan first falls below $398000 after how many months?"
• 2023 NHT Exam 2 Q6d.i — "Determine the balance halfway through the 25-year term of the loan. Round your answer to the nearest cent."

Interest paid

• 2016 Exam 2 Q7a.ii (1 mark) — "What is the total interest that Ken will have paid after 12 repayments?" Report: "Repayments − reduction in the principal = 12 × 800 − (70 000 − 65 076.22) = 9600 − 4923.78 = 4676.22. A common incorrect answer was $4923.78, which is the reduction in the principal over the year. This failed to take into account the $9600 total of repayments made in the year." Only 26%.
• 2019 Exam 2 Q9b (1 mark) — "What is the total interest Phil will have paid after three years? Round your answer to the nearest cent." Report: "Paid = 78 × $1704.03 = $132 914.34; The principal reduced by $350 000 − $262 332.33 = $87 667.67; Interest = $132 914.34 − $87 667.67". Only 19%.
• 2018 Exam 2 Q6b.ii (2 marks) — "How much interest would Julie's annuity earn in the second year of investment? Round your answer to the nearest cent." Report shows the four-line method (balance start / balance end / balance reduction / payments in year). 11% full marks.
• 2024 Exam 1 Q21 — report: "Total repaid = 2228.40 × 5 × 12 = 133 704; Interest paid = Amount repaid − amount borrowed = 133 704 − 121 000 = 12 704."
• NHT 2024 Exam 2 Q9b (2 marks) — "Determine the total amount of interest Cleo paid in the first two years of the loan. Round your answer to the nearest cent." (Repayment changed between year 1 and year 2; answer $6334.11.)
• NHT 2026 Exam 2 Q8a.ii — "Determine the total interest earned in the first year of the annuity. Round your answer to the nearest cent."

Number of payments to repay

• 2020 Exam 2 Q11 (2 marks) — "From this point on, how many more monthly repayments will Samuel make to fully repay the loan?" Answer 150. 19% full marks; a method mark was available for finding the repayment of $2400.
• 2022 Exam 1 Q19 — "With a small change to the final payment, the loan is expected to be repaid in full in A. 25 years …".
• 2024 Exam 1 Q23 — three-step solver method (rate from recurrence → annual rate → solve for N = 107.99992 ≈ 108 months ≈ 9 years). 54% correct.
• 2025 Exam 2 Q7d (1 mark) — "The last payment required to fully repay the loan is $15730.71, correct to the nearest cent. How many payments of $15730.88 did Declan make before this final payment?" Answer 59 ("60 payments in total, therefore 59 payments … before the final payment"). 41% correct.
• 2025 Exam 1 Q24 — two-phase annuity; "The total number of years for which Dennis receives payments is closest to". 52% correct.
• NHT 2026 Exam 1 Q24 — loan restructured after five years; "After the restructure, the number of monthly repayments required to fully repay the loan is".

Final / adjusted last repayment ("balloon"-type)

VCAA does not use the word "balloon" anywhere in the corpus. The examined equivalents are:

• 2018 Exam 1 Q22 — "Adam knows that the loan cannot be exactly repaid with 60 repayments of $3200. To solve this problem, Adam will make 59 repayments of $3200. He will then adjust the value of the final repayment so that the loan is fully repaid with the 60th repayment. The value of the 60th repayment will be closest to". Report: "to result in a future value of −$368.12. The negative indicates that this amount must still be paid back to the bank and added to the regular payment of $3200 to give $3568.12. Students who chose option D did not add the interest that was required to be paid with the final payment."
• 2019 Exam 1 Q23 — "Immediately after the 59th repayment is made, Joseph still owes $995.49. The value of his final repayment, to the nearest cent, will be". Report: one-period solver, N=1, PV=995.49, FV=0 → PMT = −$1001.71.
• 2021 Exam 2 Q8c (1 mark) — "For the loan to be fully repaid, to the nearest cent, Sienna's final repayment will be a larger amount. Determine this final repayment amount. Round your answer to the nearest cent." Answer $1198.59; "A common incorrect response was $1197.39. This was the balance before the last payment but does not include the interest to be added to the outstanding balance for the last payment." Only 15%.
• 2022 Exam 2 Q8c (1 mark) — "The final repayment that is required to fully pay off the loan is smaller than all other repayments by an amount less than one dollar. Determine this small amount in dollars, rounded to the nearest cent." Answer $0.15. 23% correct; "some students did not clearly indicate the small amount less than one dollar as asked."
• 2023 Exam 1 Q22 — "A final repayment that will fully repay the loan to the nearest cent is". Report gives four steps (rate → N=300 → balance before final payment → final payment).
• 2023 Exam 2 Q5c (1 mark) — "The final repayment required will differ slightly from all the earlier repayments of $1515.18. Determine the value of the final repayment. Round your answer to the nearest cent."
• 2024 Exam 2 Q8 (2 marks) — "the final repayment amount being slightly different from all the other repayments". Report warns: "Some students incorrectly used 289 payments to find a final payment of $4.20 (not slightly different) and a total cost of $885 633.64."
• NHT 2024 Exam 2 Q7e — "The final monthly repayment required to fully repay the loan to the nearest cent will be slightly higher than all previous payments."
• NHT 2026 Exam 2 Q5d — "The final repayment required by Timmy to fully repay the loan is slightly different from all other repayments. Determine this final repayment. Round your answer to the nearest dollar."

Lump-sum reduction

• 2016 Exam 2 Q7b (2 marks) — "After three years, Ken will make a lump sum payment of $L in order to reduce the balance of his loan. This lump sum payment will ensure that Ken's loan is fully repaid in a further three years. … What is the value of Ken's lump sum payment, $L? Round your answer to the nearest dollar." Report enumerates three stages. 8% full marks — the hardest recursion item of 2016.
• 2020 Exam 1 Q30 / Sample Q24 — "After 10 years, he made a one-off extra payment of $10000 to the account … The sum of money Hector originally invested is closest to." Report gives a three-step backward solver chain.
• 2021 Exam 2 Q9b (1 mark) — "Sienna will add an extra one-off amount into the annuity at this time. Determine the value of this extra amount." Answer $7039.20. 8% correct — the lowest-scoring item in the 2021 Core.

Changing conditions mid-loan

Nearly every year's hardest item. See 2017 Exam 1 Q24 (new interest rate after 10 years), 2018 Exam 1 Q24, 2019 Exam 2 Q9c, 2021 Exam 1 Q24, 2021 Exam 2 Q9b, 2022 Exam 2 Q8d ("Using a different multiplication factor (other than 1.002) the loan would be fully repaid one year sooner. Determine this multiplication factor. Round your answer to four decimal places." — 19% correct), 2023 Exam 2 Q7c, 2024 Exam 1 Q24, NHT 2024 Exam 2 Q9b, NHT 2026 Exam 1 Q24.

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1.6 Annuities and perpetuities

Annuity: payment, principal, time

• 2016 Exam 1 Q18 — "The value of an annuity, Vn, after n monthly payments of $555 have been made, can be determined using the recurrence relation V0 = 100000, Vn+1 = 1.0025Vn − 555."
• 2016 Exam 1 Q24 — the classic "two balances" annuity: "The balance of the annuity will be $130784.93 after five years. The balance … will be $66992.27 after 10 years. The monthly payment that Mai receives from the annuity is closest to". Report: students had to "recognise that the value of the annuity after 5 years could be treated as the start of a new annuity … while the value … after 10 years … could be regarded as the future value" and "correctly apply a sign convention appropriate to an annuity." This was the flagged weakness of the 2016 Exam 1.
• 2019 Exam 2 Q8 (4 marks) — A0 = 200000, An+1 = 1.0035An − 3700: a. payment ($3700, 72%); b. show that the rate is 4.2% (30%); c. "What is the balance of the annuity when it first falls below the monthly payment amount? Round your answer to the nearest cent." ($92.15, 36%); d. perpetuity payment ($700, 40%).
• 2020 Exam 2 Q10 (3 marks) — A0 = 500000, An+1 = kAn − 2000: balance after two months if k = 1.0024; annual rate; "For what value of k would this investment act as a simple perpetuity?" (1.004, 36%).
• 2023 NHT Exam 2 Q8a — "Determine the amount of money Ramon is planning to invest. Round your answer to the nearest cent."
• NHT 2026 Exam 2 Q8b — "Jane invests a principal amount, V0, in an annuity that provides a quarterly payment of $7600 for a total of 10 years … Write this recurrence relation in terms of V0, Vn+1 and Vn, with V0 rounded to the nearest cent."
• 2025 Exam 2 Q10 (2 marks) — "Halfway through the 10-year annuity, Declan writes a recurrence relation … Complete the recurrence relation below in terms of D0, Dn+1 and Dn … D0 = □, Dn+1 = 1.016 × Dn + □". Answer D0 = 376 159.43, … + (−22 126.27). 11% full marks; "Some students found the payment correctly from the calculator but answered as positive 22 216.27."

Perpetuities

Reports call this out as a persistent weakness across many years. - 2016 Exam 1 Q21 — choose the graph of "$80000 in a perpetuity that will provide $4000 per year". - 2017 Exam 2 Q7a — "What monthly payment will Alex receive from this investment?" ($1560). "Some students gave the annual payment of $18 720, while others knew the correct method but rounded 5.2 to two decimal places before multiplying by $360 000, giving $1 548." 50% correct. - 2018 Exam 2 Q6a — "Julie could invest the $492800 in a perpetuity. She would then receive $887.04 each fortnight … At what annual percentage rate is interest earned by this perpetuity?" Answer 4.68%. "A common error was 0.18%, which did not account for the fortnightly payments. Perpetuities was an area that requires improvement for many students." 41% correct. - 2021 Exam 2 Q6 (3 marks) — a. total received after one year; b. "Write down the value of the perpetuity after Sienna has received one year of monthly payments" ($420 000 — 47%; "The definition of a perpetuity continues to be misunderstood by many students"); c. complete S0 = □, Sn+1 = □ × Sn − 1890 (38%; "A factor of 1.054 was a common incorrect response"). - 2022 Exam 2 Q7d — "If Pina had invested the original $540000 annuity as a simple perpetuity, what monthly payment would she have drawn?" ($1350). "Many students were unable to provide the definition of a perpetuity. The annuity payment of $5214.28 was the most common mistake." - 2024 Exam 2 Q7d — 0.003 × 300 000 → $900. PL transcript: "Perpetuities remain an area of concern for students who do not understand that these are a special case of an annuity that lasts indefinitely." - 2025 Exam 1 Q21 — "When will the sum of all annual payments to Donald first exceed $250000?" Report: "Annual payment (5% of $250 000) = $12 500; Years required to equal $250 000 = 20; One more year required to first exceed $250 000." 69% correct. - NHT 2022 Exam 1 Q22 — "V0 = a, Vn+1 = 1.004Vn − b. Which of the following could not be values for a and b?" - NHT 2023 Exam 1 Q22 — "If James receives a regular monthly payment of $1440, then the balance of the perpetuity after one year of monthly payments is" (unchanged: $480 000). - NHT 2026 Exam 1 Q18 — "The payment received by Nerida from this investment in the fifth month is" (constant $1800). - Legacy: 2015 Exam 2 Module 4 Q3 — "a. Determine the minimum amount they must invest in the perpetuity to fund the scholarship. b. For how many years will they be able to provide the scholarship?" Answer: "It will last forever. Many students did not know that perpetuities pay out only the interest earned, while the principal remains unchanged. The most common incorrect answer was 12500/460 ≈ 27 years."

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1.7 Amortisation tables

Reading / interpreting

• 2016 Exam 1 Q22 — three-line table; "The amount of payment number 2 that goes towards reducing the principal of the loan is".
• 2018 Exam 1 Q23 — five lines; "The interest rate for this loan changed immediately before repayment number 28. This change in interest rate is best described as". Report shows both monthly rates and the ×12 conversion.
• 2019 Exam 1 Q20 — "The annual interest rate for this loan is 3.6%. Interest is calculated immediately before each payment. For this loan, the repayments are made A. weekly … E. yearly." (Answer: monthly, since 0.3%/period.)
• 2021 Exam 1 Q18 & Q19 — principal reduction for payment 3; total number of years of payments.
• NHT 2022 Exam 1 Q21 — "Mustafa made his first payment on 1 January 2021. He made his second payment on A. 8 January 2021 … E. 1 January 2022" (deduce the period from the table).
• NHT 2023 Exam 1 Q18 & Q19 — balance after payment 3; interest rate per annum.
• 2024 Exam 1 Q22 — report: "Interest rate = …; Interest amount payment 2 = 238 218.95 × 0.004 = 952.8758 ≈ 952.88; Principal reduction = payment − interest = 2741.05 − 952.88 = 1788.17."

Completing missing entries

• 2017 Exam 1 Q23 — annuity-investment table with a varying payment: "The interest rate for this investment remains constant, but the payment value may vary. … The value of payment number 20 is closest to". Report: "Many students seemed to assume that the payment value would be constant at $100 (option B), despite the question stating that the payment may vary. The solution required four steps."
• 2020 Exam 2 Q8 (3 marks) — "a. Using the values in the amortisation table i. calculate the principal reduction associated with payment number 3 ii. calculate the balance of the loan after payment number 4 is made. Round your answer to the nearest cent. b. Let Sn be the balance of Samuel's loan after n months. Write down a recurrence relation, in terms of S0, Sn+1 and Sn, that could be used to model the month-to-month balance of the loan." (8a.ii 39%, 8b 30%.)
• 2023 Exam 2 Q6b (1 mark) — "Using the values in the table, complete the next line of the amortisation table. Write your answers in the spaces provided in the table below. Round all values to the nearest cent."
• 2025 Exam 2 Q7c (1 mark) — "The interest rate on Declan's loan is 4.2% per annum, compounding monthly. Using the values in the table, complete the table below. Round all values to the nearest cent." Answers 2885.55 | 12 845.33 | 811 598.26. Guide: "Designated rounding question. Accept only this answer." Report: "Students were instructed to use the values in the table. Using a finance solver resulted in a different balance to the nearest cent which was not appropriate." 40% correct.
• NHT 2024 Exam 2 Q7d — "Complete the next line in the amortisation table. Write your answers in the spaces provided in the table below."
• NHT 2026 Exam 2 Q7 (2 marks) — "After one year, Timmy changes to making interest-only repayments. Complete the next line in the amortisation table below, rounding all values to the nearest cent." Guide: "Correct interest OR Any two values correct / ALL four values correct."
• NHT 2021 Exam 2 Q5c (2 marks) — "Write down the values of P, Q and R, rounded to the nearest cent, in the boxes provided below."
• 2022 Exam 2 Q7a/7b — "What is the value of payment number 3?" then "Calculate the interest associated with payment number 3. Round your answer to the nearest cent." Report: "Rounding errors occurred by those students who calculated their own interest rate rather than using the information given in the question."

Deriving the recurrence from the table

Standard follow-up: 2020 Q8b, 2022 Q7c, 2023 Q6c, NHT 2022 Q7c.

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1.8 Interest-only loans

Small but reliably present; usually 1 mark or one MCQ. - 2017 Exam 1 Q19 & Q20 (paired stem) — "Each month, Shirley will pay only the interest charged for that month. … After three years, the amount that Shirley will owe is" (unchanged: $225 000) / "A recurrence relation that models the value of Vn is" (V0 = 225000, Vn+1 = 1.003Vn − 675). Report explains both the algebraic route and the "constant sequence" shortcut: "Option D was the only recurrence relation that resulted in this constant sequence." - 2016 Exam 1 Q20 option E — "an interest-only loan of $10000." - 2021 Exam 1 Q24 — "Bob decided, however, to make interest-only repayments for the first two years. After these two years the interest rate changed. Bob was still able to pay off the loan in the 20 years by repaying the scheduled amount each month. The interest rate, per annum, for the final 18 years of the loan was closest to". - NHT 2018 Exam 2 Q9a — "For the first year of this loan, Andrew will make interest-only repayments each month. What is the value of each interest-only repayment?" ($107.50). - 2023 Exam 2 Q7b.i — "Determine the value of d in the recurrence relation if i. Arthur makes interest-only repayments". - NHT 2021 Exam 1 Q17 option E — "a reducing balance loan with interest-only repayments." - NHT 2026 Exam 2 Q7 — amortisation line for the switch to interest-only. - Legacy: 2012 Exam 2 Module 4 Q3c.ii — "After the first 12 months, only the interest on the loan is paid each month. Determine the monthly interest payment, correct to the nearest cent."

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1.9 Using the finance solver / TVM — sign conventions

VCAA examination reports print the solver input block as the model answer. The canonical layout (from Documents_exams_mathematics_2016_FM2_examrep16.txt Q7a.i) is:

N     = 12
I%    = 6.9
PV    = 70 000
PMT   = −800
FV    = −65 076.219...
P/Y = C/Y = 12

Sign convention rules the reports state or demonstrate: - Loan taken out: PV positive (money received), PMT negative, FV negative (still owed). 2016 Q7a.i; 2019 Exam 1 Q23; 2022 Exam 2 Q8c. - Investment/annuity: PV negative (money paid out), FV positive; payments received are positive PMT, payments made into the investment are negative PMT. 2018 Exam 1 Q24 report: "PV = −307 794.50 (Entered as negative here. It represents the new principal value invested.) … PMT = ? … FV = 600 000 … to result in a PMT value of −1854.05, meaning outgoing payments are close to $1854." - Explicit warning, 2019 Exam 2 Q9a: "$7954.54 was a common incorrect answer obtained by entering the FV in technology as a positive rather than as −262 332.33. This very large fortnightly repayment should have been a signal that something had been entered incorrectly." - Explicit statement, 2018 Exam 1 Q24 report: "Understanding of the sign convention for the finance solver is very important, as is the careful tracking of values used in subsequent calculations." - P/Y and C/Y must match the compounding period, and N is in periods, not years: reports write N = 5 × 12 = 60 (2016 Exam 1 Q24), N = 78 (2019 Exam 2 Q9a, fortnightly for 3 years), N = 2 × 12 (2024 Exam 1 Q24). - Multi-stage chaining (the standard 2-mark / hard-MCQ structure): find FV of stage 1, carry it in as the PV of stage 2, keeping full decimal accuracy. 2016 Q7b; 2017 Exam 2 Q7b; 2018 Exam 1 Q24; 2020 Exam 1 Q30; 2021 Exam 2 Q9b; 2022 Exam 2 Q8c; 2024 Exam 1 Q24; NHT 2024 Exam 2 Q9b. - P/Y = C/Y = 1 with a per-period rate is an acceptable alternative the reports themselves use: 2021 Exam 1 Q19 (I% = 4, P/Y = 1, C/Y = 1); 2025 NHT Exam 1 Q22 (I% = 0.3, P/Y = 1, C/Y = 1). - Caution against CAS syntax in the response, 2021 Exam 2 Q8b report: "Students must be aware that any response including CAS syntax/notation cannot attract full marks. In the instance of 1-mark questions (particularly 'show that' questions such as this one), the student may not be awarded the mark." - Method marks for solver entries, 2016 Exam 2 Q7b report: "Correct tables of input values for the financial solver may have illustrated working out to qualify for [a method mark]." And 2017 Exam 2 general comments: "the finance solver are examples by which that understanding may be shown."

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1.10 "Show that" questions using recursion step-by-step

This is the single most heavily policed answer-form in the topic. See §2.2 for the verbatim rules. Catalogue of occurrences:

Non-recursive "show that" in this topic (arithmetic proof of a rate or amount):

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1.11 Effective interest rate comparisons

Formula-sheet supported. Two distinct forms.

(a) Compute / compare effective rates - 2018 Exam 1 Q19 — five loans with different nominal rates and different compounding periods; "When fully repaid, the loan that will cost Daniel the least amount of money is". Report: "This question required an understanding that both the interest rate and compounding period in combination determine the overall cost of the loan. Students who selected option A seemed to simply select the lowest interest rate without considering the compounding period." - 2019 Exam 1 Q24 — "The difference between the rate of interest per annum used by her bank and the effective annual rate of interest for Millie's investment is closest to". Two-stage: solve for the nominal rate, then apply the formula. - 2020 Exam 1 Q28 — "The nominal interest rate for a loan is 8% per annum. When rounded to two decimal places, the effective interest rate for this loan is not A. 8.33% … daily … E. 8.24% … quarterly." - 2021 Exam 1 Q21 — "The effective annual interest rate for this investment, rounded to two decimal places, is". - 2022 Exam 1 Q21 — four true/false statements about nominal vs effective rates; "How many of these four statements are true?" Report: "Statements 1, 2 and 3 are true." Only 33% correct (45% chose C). - 2023 Exam 1 Q23 — read the effective rate off a loan-balance graph (two-step: quarterly interest from the graph → annual rate → effective rate). 37% correct. - 2025 Exam 1 Q23 — the reverse: effective rate given (4.51%), find the nominal rate, then the interest earned. Report: "Step 1. The nominal interest rate = nom(4.51,26) = 4.415% p.a." 41% correct. - NHT 2024 Exam 1 Q20, NHT 2023 Exam 1 Q23, NHT 2022 Exam 1 Q23, Sample Exam 1 Q23 — straightforward effective-rate computations. - NHT 2026 Exam 1 Q21 — pure formula recognition: "The effective rate of interest for Josh's investment account is equal to A. [(1 + 4.5/100)⁴ − 1] × 100% B. [(1 + 4.5/400)⁴ − 1] × 100% …". Tests whether students know r/(100n) not r/100.

(b) Explain the gap between nominal and effective - 2024 Exam 2 Q6b (1 mark) — Answer: "It does not take into account the fortnightly compounding." Only 26%. Report: "This question was not answered well. Many students had some understanding of the effective interest rate and could even define it, but could not explain in simple terms why the nominal rate was lower." Assessor PL: "This student was awarded the mark for including that the effective is based on compounding periods. … This student didn't include reference to the compounding periods." - Legacy 2013 Exam 2 Module 4 Q3c — "Explain why the effective interest rate per annum is higher than the flat interest rate per annum." Report answer: "Simple interest does not take into account reductions in the outstanding balance as repayments are made. … Each monthly repayment reduces the balance of the loan and the regular $50 interest payment becomes a greater percentage of this remaining balance. … The effective rate is approximately double the flat rate and represents an average actual rate paid in the life of the loan." - Legacy formula (2006–2015 sheet only): effective rate of interest ≈ 2n/(n+1) × flat rate, used in 2013 Exam 2 Mod 4 Q3b: "Effective rate = 2×24/(24+1) × 8 = 15.36".

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1.12 "Explain why" / "name this" / "what does this represent" questions

These are 1-mark descriptive items. VCAA marks them strictly on the presence of the key idea.

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1.13 LEGACY (pre-2016): Number patterns module — extra practice source

Content overlaps modern recursion but adds sums of sequences and steady-state analysis which are no longer examined.

• Arithmetic sequence + difference equation, 2015 Exam 2 Module 1 Q1 (5 marks) — "The weight of the capsicums … forms the terms of an arithmetic sequence, as shown below. 2000, 2150, 2300 ... a. Write down a calculation that shows that the common difference for this sequence is 150. b. How many kilograms … in week 5? c. How many kilograms … in total … in the first eight weeks? d. In which week will the total weight of the harvest first exceed 14 000 kg? e. The weight of the capsicums, Cn, picked in week n, is modelled by a difference equation. Write down the rule for this difference equation in the box provided below. Cn + 1 = □ C1 = 2000" — note the legacy 1-based index and answer-second layout. Marks distribution: 25% scored 8/8.
• Geometric sequence, 2015 Exam 2 Module 1 Q2 (7 marks) — plot a term on a graph; "At the beginning of which week will the number of whiteflies first be expected to exceed 500 per square metre?"; "What is the percentage increase … from the beginning of any week to the beginning of the next week?" (60% — "A common incorrect answer was 160%"); then a mixed recurrence Wn+1 = 1.6 Wn − k.
• Exam 1 legacy MCQs, 2013 Module 1 — Q4 "The total vertical distance … that the balloon rises in the first 10 minutes" (geometric sum); Q5 "A sequence is generated by the difference equation tn+1 = 2 × tn, t1 = 1. The nth term of this sequence is" (tn = 1 × 2^(n−1)); Q6 arithmetic-sum word problem; Q7 "Which one of these sequences could not include 2 as a term?"; Q8 "Sn+1 = 0.2 × Sn + 15, S1 = 10. The maximum hourly rate of pay that the worker can earn in this job is closest to" (steady state 18.75).
• Other rich legacy items: Documents_exams_mathematics_2008furmath2-w.txt L293–370 (cows/deer difference-equation modelling, "Write a difference equation for Dn in terms of Dn−1"); ...2009furmath2-w.txt L294–311 ("Show that the sequence generated by this difference equation is not arithmetic"; "What does the difference equation indicate about the long-term future of this variety concert?"); ...2010furmath1-w.txt L452 (Fibonacci-style tn = tn−1 + tn−2).

1.14 LEGACY (pre-2016): Business-related mathematics module — extra practice source

Everything in §1.2–§1.8 has a legacy analogue, plus three topics dropped in 2016: GST/percentage change, hire purchase, and deposits/instalments.

• 2015 Exam 2 Module 4 Q1 — "The business charges customers $88 per hour. The $88 per hour includes a 10% goods and services tax (GST). a. Calculate the amount of GST included in the $88 hourly rate." Report: "A common incorrect answer was 10% of 88 = $8.80."
• 2015 Exam 2 Module 4 Q2 — flat-rate "show that" (§1.10), time to reach $550, and "Find the annual percentage rate by which the value of this recording equipment depreciated. Write your answer correct to two decimal places." (13.11%).
• 2013 Exam 2 Module 4 Q3 — hire purchase: "A flat interest rate of 8% per annum was charged. He will fully repay the principal and the interest in 24 equal monthly instalments. a. Determine the monthly instalment … b. Find the effective rate of interest per annum … c. Explain why the effective interest rate per annum is higher than the flat interest rate per annum. d. … reducing balance … Determine the value of the bike five years after it was purchased. Write your answer, correct to the nearest dollar." Report warns: "Students often confuse simple (flat) interest with compound interest calculations. Many students inappropriately treated this as a reducing balance problem. Students should understand that the term 'flat interest rate' refers to simple interest."
• 2012 Exam 2 Module 4 Q2 — unit cost + flat rate equivalence + "After how many years will equipment be written off with a depreciated value of $0?" + reducing balance after ten years.
• 2012 Exam 2 Module 4 Q4 — perpetuity: annual rate from quarterly payment; balance after 20 payments (unchanged); reverse-solve an annuity principal.
• 2009 Exam 2 Module 4 Q4c — "After 4 years, which method, flat rate depreciation or reducing balance depreciation, will give the greater depreciation? Write down the greater depreciation amount in dollars correct to the nearest cent."
• 2013 Exam 1 Module 4 MCQs — percentage discount; successive +5%/−5% ("Which one of the following calculations will give the closing price"); "Which one of the following calculations will give the total interest earned, in dollars, by this investment?" (10 000 × 1.02²⁰ − 10 000); flat rate from deposit + instalments; perpetuity balance after 10 quarterly payments; reverse percentage growth.

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2. THE VCAA WORDING BANK

2.1 "Write down a recurrence relation, in terms of …"

Canonical template (2016–2020):

"Let [variable]n be the [quantity] after n [periods].
Write down a recurrence relation, in terms of [V]0, [V]n+1 and [V]n, that [could be used to model / models / can be used to determine] the [quantity]."

Template drift by year (all corpus-verified):

THE REQUIRED TWO-PART ANSWER — verbatim examiner rules

"A recurrence relation has the initial value written first, followed by the recurrence rule.
Common errors included:
- failure to include the initial value, C0
- writing the initial value as Cn, not C0
- using different symbols for different parts of the recurrence relation, e.g. V0 = 38 000, Cn+1 = Cn − 2750
- using a rule for the nth term Cn = 38000 − 2750n instead of the recurrence rule."
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q6b

"Many students did not write a recurrence relation in the required form.
A recurrence relation has the initial value written first.
The name of the variable needed to be consistent in the recurrence relation.
Some students did not recognise the difference between the recurrence relation above and the rule An = 428 × 1.015^n."
— Documents_exams_mathematics_2017_fm2_examrep17.txt, Q6b

"Responses generally used the notation well, but many wrote a recurrence relation reflective of reducing balance depreciation rather than flat rate.
A few wrote both a recurrence relation and a rule with Vn in terms of n which could not be accepted."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, Q7d

"Notation was a problem for some students. A common error was to finish the recurrence relation with 12500n."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, Q6c

"This question was not answered well. The multiplying factor of 1.0025 was frequently incorrect and often the 5214.28 was added. Notation was a problem, with many students using a mixture of 'P notation' and 'V notation'."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, Q7c

"Recurrence relations must have two components to be complete. The initial value, V0, and the relation itself. … Whilst correct use of notation is considered to be important, the way this student used the n+1 subscript did not preclude them from being awarded the mark. … The mark was not awarded to this student. It was common to see the n added at the end of an otherwise correct answer."
— 2025-04_General_Maths_Examination_2.txt (2024 Chief Assessor PL transcript), Q5c

"Some students incorrectly included an '[n]' on the end of the recurrence relation."
— 2025-03_2024generalmaths2-report.txt, Q5c

"The multiplying factor was sometimes given as 0.003, 0.997 or 1.036. A few students mixed up Sn and Vn."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, Q8b

"The question was done reasonably well, but some responses gave B0 as 350 000, or R as 1.048, and added rather than subtracted the 3517.28."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, Q9c

Marking-guide statements of the two-part requirement:

"V0 = 40 000 Vn+1 = Vn – 8000 | A1 | BOTH parts must be correct"
— 2025-11_2025-GeneralMaths2-assessment-guide.txt, Q8a.i
"V0 = 300 000 Vn+1 = 1.0025 × Vn – 1570.58 | A1 | Both parts must be correct"
— 2025-06_2025NHT-GeneralMath2-assessment-guide.txt, Q7c
"B0 = 42 623.31 Bn+1 = 1.0045 × Bn – 2470.48 | A1A1 | Any two values correct / Fully correct recurrence relation" (2-mark version)
— 2025-06_2025NHT-GeneralMath2-assessment-guide.txt, Q8b

MANDATORY ELEMENTS checklist for full marks: 1. Initial value written first, subscript 0 (not n, not 1). 2. Recurrence step with subscript n+1 on the left and n on the right. 3. Same variable letter throughout and the letter the question names. 4. No n in the recurrence step (that belongs only in the rule form). 5. Correct sign on d (subtract for loans/annuities/depreciation; add for savings plans). 6. Correct multiplying factor for the stated compounding period (per-period, not per-annum). 7. Do not also supply the closed-form rule — a competing rule invalidates the response.

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2.2 "Use recursion to show that …" — the reports are emphatic

Wording variants across years: - "Use recursion to write down calculations that show that the amount of money in Ken's savings account after two years, V2, will be $16224." (2016 Exam 2 Q5b) - "Recursion can be used to calculate the value of the van after two years. Complete the calculations below by writing the appropriate numbers in the boxes provided." (2017 Exam 2 Q5a) - "Write down a calculation to show that the balance in the account after one month, V1, is $12074.40" (2018 Exam 2 Q4b.i) - "Use recursion to show that the value of the tools after two years, V2, is $48600." (2019 Exam 2 Q7a; NHT 2017 Q5b; NHT 2021 Q6a) - "Showing recursive calculations, determine the value of the machine, in dollars, after two years." (2020 Q7b; 2021 Q9a; 2022 Q8a; 2023 Q5b; NHT 2022 Q5b; NHT 2023 Q6b; Sample Q6b) - "Show, using a recursive calculation, that the balance of the loan after one month is $598664.07" (NHT 2026 Q5b)

Note: the phrase "Recursively calculate …" does not occur anywhere in this corpus. The instruction is always one of the six forms above.

THE RULES — quoted verbatim

2016 (and NHT 2017) — the fullest statement:

"V0 = 15 000
V1 = 1.04 × 15 000 = 15 600
V2 = 1.04 × 15 600 = 16 224

'Using recursion' begins with writing the initial value (i.e. V0 = 15 000)

Then the first calculation using the recurrence rule must be shown and the answer labelled.
(V1 = 1.04 × 15 000 = 15 600 in this case)

Subsequent calculation(s) must then show the use of the prior answer to calculate the next value.
(V2 = 1.04 × 15 600 = 16 224 in this case)

Some students used only the explicit rule Vn = R^n V0 to write V2 = 1.04² × 15 600 = 16 224, but this was not using recursion as required."
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q5b
(verbatim repeat in Documents_exams_mathematics_2017_nht_fm2nht_examrep17.txt, Q5b)

2019 — general comments section, the canonical statement:

"In Core question 7a. students were asked to use recursion to show that the value of the tools after two years, V2 = $48 600. This means students must show each iteration by writing down the relevant calculation:
V1 = 60 000 x 0.9 = 54 000
V2 = 54 000 x 0.9 = 48 600

To state V1 = 54 000 without showing how this was obtained does not address the instruction to 'show'.

Using V2 = 60 000 x 0.9² = 48 600 does not address the use recursion instruction."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, general comments

2020 — with the explicit non-example:

"In Core Question 7b. students were asked to show recursive calculations that would determine the value of the machine after two years. This meant students needed to show each iteration by writing down the relevant calculation:
V1 = 120 000 – 15 000 = 105 000
V2 = 105 000 – 15 000 = 90 000
The final answer on its own, or a statement such as V2 = 105 000 – (2 x 15 000) = 90 000, would not satisfy the requirements of this question."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, general comments

2022 — the shortest, bluntest version:

"Core Question 8a. required students to 'show recursive calculations'. The mark can only be obtained by showing the step-by-step approach. An answer on its own is not sufficient to attract the marks."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, general comments

2024 — marking guide instruction:

"E0 = 300 000
E1 = 1.003 × 300 000.00 – 2159.41 = 298 740.59
E2 = 1.003 × 298 740.59 – 2159.41 = 297 477.40
| A1 | SHOW THAT question. Show BOTH calculations and final answer is correct"
— 2025-04_2024generalmathematics2-assessment-guide.txt, Q7a

"Note that an answer of $297 477.4 is not correct to the nearest cent."
— 2025-03_2024generalmaths2-report.txt, Q7a

FULL-MARK RESPONSE STRUCTURE (mandatory elements): 1. Write the initial value with its subscript: V0 = 60 000. 2. Write the first iteration as a complete calculation with the operation visible and the result labelled: V1 = 0.9 × 60 000 = 54 000. 3. Write each subsequent iteration using the previous answer: V2 = 0.9 × 54 000 = 48 600. 4. Never use V0 × R^n, V0 − n × d, or a finance-solver screen. 5. The final value must equal the given number, correctly rounded (and to the nearest cent if a currency answer).

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2.3 The generic "show that" rule (applies to non-recursive show-that items)

"'Show that' questions give the answer to a basic and appropriate calculation, and students must write that calculation. The given number in a 'show that' question is sometimes needed in a following question. With this number, students can attempt the following question, even if they cannot complete the 'show that' question.
The given number in a 'show that' question is not to be used in a calculation; it must be the result of a calculation. The following calculations use the 2750 to show a different result than what is required and were not acceptable for this question:
- 38 000 − 8 × 2750 = 16 000 — shows how to find the depreciated value after eight years
- 16 000 + 8 × 2750 = 38 000 — shows how to find the initial value"
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q6a (near-identical text in Documents_exams_mathematics_2015_FM2_examrep15.txt)

"In the Recursion and financial modelling module Question 8b. students were asked to show that the annual interest rate is 4.2% from the given recurrence relation. Stating that 1 + r/1200 = 1.0035 and r = 4.2% amounts to recognising the well-known interest rate rule and then stating the conclusion. To show that the result is the case, some calculation must be provided:
r/1200 = 0.0035
r = 1200 x 0.0035 = 4.2%
… In each of these cases the result must be the conclusion of the reasoning process. Showing the process is the basis of achieving the mark."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, general comments

"Responses needed to show the calculation using basic arithmetic that resulted in an answer of 4.2%, not use the 4.2% and attempt to verify it."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, Q8b

"Questions that were not well answered often involved the instruction to 'show that' a particular answer could be obtained. Students must work towards the given result with all relevant steps shown. When a student started with the answer given, the mark could not be awarded. For example, in core Question 8b., students were asked to show that the compound interest rate for a loan was 2.6% per annum. A student who worked back from this value to find the factor of 1.001 in the recurrence relation could not be given the mark."
— Documents_exams_mathematics_2021_2021furthermath2-report.txt, general comments

"In other words, if a student starts with the answer they have been asked to show, they will not be eligible for the mark."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, general comments

"A question that asks to 'show that' a particular answer could be obtained was a problem for many students. Make sure when doing a 'show that' question, any equations or values used to 'show' the answer are copied correctly. Students should recall that they cannot simply substitute a value or use the fact they are attempting to 'show' as part of this process."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, general comments

"Does the answer match the value provided in a 'show that' question?" (in the list of self-checks students should run)
— 2025-03_2024generalmaths2-report.txt, general comments

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2.4 Rounding instructions and how examiners penalise

The instruction strings actually used (corpus counts across all files)

Also seen: "with V0 rounded to the nearest cent" (NHT 2026 Q8b), "Write your answer as a percentage" (2017 Q5b.ii), "Write your answer correct to two decimal places" (NHT 2017 Q7a; legacy 2015 Mod 4 Q2c).

Verbatim penalty rules

Nearest cent means exactly two decimal places — no further rounding:

"Currency answers rounded to the nearest cent needed to show two decimal places. Many students only wrote an answer of $22 203.7, which might arise from a calculator setting that restricts the number of displayed digits. Some students appeared to round all currency answers to the nearest five or ten cents. A common incorrect answer was $22 203.70, which had clearly been rounded from a written $22 203.664... This was assessed as a rounding error."
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q5d.ii

"In Recursion and financial modelling students are often asked to round their answer to the nearest cent. It is not appropriate to round to the nearest five cents or 10 cents in this instance."
— Documents_exams_mathematics_2017_fm2_examrep17.txt, general comments

"In the Recursion and financial modelling module students sometimes continued working from a correct answer to an incorrect answer because they rounded to the nearest 5 cents or 10 cents. For example, in Question 5b. of the Core section, the required answer was $5484.23 and some students incorrectly rounded further to $5484.20 or $5484.25."
— Documents_exams_mathematics_2018_FM2_examrep18.txt, general comments

"…students sometimes further engaged from a correct answer to an incorrect answer because they rounded to the nearest 5 c or dollar. They needed to treat the answer to the nearest cent as if it were electronic money where no further rounding occurs. For example, in Core Question 9a. the required answer was $1704.03 and some students incorrectly rounded further to $1704.05 or $1704."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, general comments

"In Recursion and financial modelling students are often instructed to answer questions to the nearest cent (which means two decimal places). Students sometimes unnecessarily changed from a correct answer to an incorrect answer because they rounded to the nearest 5c, 10c or dollar. In all questions where rounding applies, this is clearly stated in the question."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, general comments

"In Recursion and financial modelling, students are often instructed to answer questions correct to the nearest cent (two decimal places). For example, in core Question 9b., the required answer to the nearest cent was $7039.20. A student who answered $7039.2 could not be awarded the mark. This given answer actually represents one decimal accuracy of any value between $7039.15 and $7039.24."
— Documents_exams_mathematics_2021_2021furthermath2-report.txt, general comments

"In Financial Mathematics, students should only round to the nearest dollar or nearest cent when specified."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, general comments

No instruction to round ⇒ exact answer required:

"In questions where no instruction to round is given, an exact answer is required as rounding does not apply."
— 2017, 2018, 2019, 2020, 2021, 2022 Exam 2 reports (identical sentence each year)
2018 example: "in Question 2c. of the Core section the required answer was 63.8% and since rounding did not apply an answer of 64% could not be accepted."
2024 example: "in Question 1e the percentage quoted needed to be 85.7% not 86%" (2025-03_2024generalmaths2-report.txt).
2024 PL: "This is an example where no instruction to round was given in the question. In all questions of this type, the exact answer must have been given. An exact answer may be a decimal, as long as it is a terminating decimal … 20.8% was the only acceptable answer, and not, for example, 21%." (2025-04_General_Maths_Examination_2.txt)

Premature rounding:

"Students should also take care to avoid premature rounding. They should use all the decimal places in their calculator for intermediate steps and then only round the final answer to the required accuracy level."
— Documents_exams_mathematics_2017_fm2_examrep17.txt, general comments
Worked example of the penalty: 2017 Q7a — "others knew the correct method but rounded 5.2 to two decimal places before multiplying by $360 000, giving $1 548."

The one-rounding-error concession (2020–2022 Core):

"In Core questions where a maximum of one rounding error was penalised, a second rounding error would not have resulted in the loss of marks if either:
- a correct calculation was shown prior to the final rounded answer
- additional correct decimal places could be seen earlier in the response or in the final answer."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt (repeated 2021, 2022)

The 2024+ "designated rounding question" regime — the current rule:

"For a rounding error to be considered, firstly, rounding must apply to the question, then the assessor must see either a correct calculation shown prior to the final rounded answer, or additional correct decimal places. If a student rounds in a question where rounding does not apply, this is considered an absolute error, and no answer mark is awarded. … The third rounding question type involved rounding to the nearest cent in relation to the recursion and financial modelling area of study. In addition to the calculations, the answer had to be $297,477 and 40 cents."
— 2025-04_General_Maths_Examination_2.txt, 2024 Chief Assessor PL

Marking-guide flags in the current era: - *** Rounding to nearest cent applies / *** Rounding to 2 dp applies / *** Rounding to 4 dp applies / *** Whole number rounding applies — 2025-11_2025-GeneralMaths2-assessment-guide.txt, 2025-06_2025NHT-GeneralMath2-assessment-guide.txt. - "Designated rounding question. Accept only this answer" — 2025 Q7c. - " Rounding to nearest cent applies, *providing Finance Solver working shown and additional decimal places shown" — 2024 Q8. - "H1 – Must see calculation using 9b.i answer *** Rounding to 4 dp applies" — 2025 Q9b.ii (a consequential/H mark).

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2.5 Interest rate wording and the conversion trap

Stem phrasings (frequency across corpus):

Other recurring stem sentences: - "Interest on this loan is charged at the rate of X% per annum, compounding [period]." (2016 Q7; 2023 Q5) - "Interest is calculated monthly and [name] makes monthly repayments." (2020 Q8; 2025 Q7; NHT 2021 Q5) - "Interest is calculated and paid monthly." (2017 Q7a perpetuity; NHT 2026 Q18) - "Interest is calculated immediately before each payment." (2019 Exam 1 Q20; NHT 2022 Exam 1 Q21) - "Immediately after the interest has been added/calculated each month, [name] deposits a further $X." (2016 Exam 1 Q23; 2017 Q7b; NHT 2017 Q7; 2021 Exam 1 Q22) - "If a customer does not pay the bill by the due date, interest is charged. Alex charges interest after the due date at the rate of 1.5% per month on the amount of an unpaid bill. The interest on this amount will compound monthly." (2017 Q6) - "Assume there are exactly 52 weeks in one year." / "Assume that there are exactly 26 fortnights in one year." (2025 Exam 1 Q22, Q23; 2025 Q9b) - "The annual interest rate for this loan is 3.6%." / "The interest rate for the annuity is 0.42% per month. Determine the interest rate per annum." (2023 Q6a)

THE CONVERSION TRAP — examiner evidence

(a) Per-period rate quoted as if annual (and vice versa).

"Many students did not read that the given interest rate was per month, so $200.25 was a common incorrect answer." — 2017 Q6a (Documents_exams_mathematics_2017_fm2_examrep17.txt)
"Many gave the annual interest rate percentage." — 2020 Q9b, where 0.3% (monthly) was wanted.
"0.24% per month was sometimes given." — 2020 Q10b, where 2.88% p.a. was wanted.
"0.2% was a common incorrect answer." — 2022 Q8b (2.4% p.a. wanted).
"Those who were incorrect often gave 0.3%, as this was the monthly rate." — 2024 Q7c.
"Some students gave the weekly interest rate rather than the annual rate, presumably through not reading the question carefully." — 2024 PL, Q5d.
"A weekly interest rate of 0.4% was a common incorrect response." — 2025-03_2024generalmaths2-report.txt, Q5d.

(b) Frequency of payment vs frequency of compounding.

"A common error was 0.18%, which did not account for the fortnightly payments." — 2018 Q6a.
"Some students gave the annual payment of $18 720." — 2017 Q7a.

(c) Rate and compounding period must be considered together.

"This question required an understanding that both the interest rate and compounding period in combination determine the overall cost of the loan." — 2018 Exam 1 Q19.

(d) The n in the rule must be the number of compounding periods.

"The growth factor per quarter was [1.0075]. As interest was compounding quarterly, the number of compounding periods in n years is equal to 4n, which is the required power." — 2020 Exam 1 Q26.

(e) Multiplying-factor derivation is expected to be shown.

"The multiplying factor is 1 + (3.6/12)/100 = 1.003. The repayment is (3.6/12)/100 × 225 000 = 675." — 2017 Exam 1 Q20 report.
2024 NHT Q7c: "Use this value in a calculation to show that the multiplication factor in the recurrence relation is 1.005."

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2.6 Amortisation table completion wording

"Using the values in the table, complete the next line of the amortisation table. Write your answers in the spaces provided in the table below. Round all values to the nearest cent." — 2023 Exam 2 Q6b

"Complete the next line in the amortisation table. Write your answers in the spaces provided in the table below." — NHT 2024 Exam 2 Q7d

"The interest rate on Declan's loan is 4.2% per annum, compounding monthly. Using the values in the table, complete the table below. Round all values to the nearest cent." — 2025 Exam 2 Q7c

"After one year, Timmy changes to making interest-only repayments. Complete the next line in the amortisation table below, rounding all values to the nearest cent." — NHT 2026 Exam 2 Q7

"Write down the values of P, Q and R, rounded to the nearest cent, in the boxes provided below." — NHT 2021 Exam 2 Q5c

MANDATORY: use the table's own numbers, not a solver.

"Students were instructed to use the values in the table. Using a finance solver resulted in a different balance to the nearest cent which was not appropriate."
— 2026-01_2025-GeneralMaths2-report.txt, Q7c

"Rounding errors occurred by those students who calculated their own interest rate rather than using the information given in the question. Rounding was required to the nearest cent."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, Q7b

Column relationships examiners expect students to state: - interest = (rate per period) × (previous balance) - principal reduction = payment − interest - new balance = previous balance − principal reduction - rate per period = interest ÷ previous balance (used to derive the rate — 2018 Exam 1 Q23; 2019 Exam 1 Q20; NHT 2023 Exam 1 Q19; NHT 2022 Exam 2 Q7a)

"Students must be familiar with the calculations that are used to create an amortisation table and be able to use these to complete questions such as this."
— Documents_exams_mathematics_2017_fm1_examrep17.txt, Q23

"The very small difference in these two values is due to the rounding of all values in the amortisation table. Either could be used to determine the interest at payment number 20."
— same, Q23

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2.7 Final / adjusted repayment wording

The method the reports prescribe (2019 Exam 1 Q23 model):

N     = 1
I%    = 7.5
PV    = 995.49        (balance outstanding after the second-last payment)
PMT   = solve   → −1001.71
FV    = 0
P/Y = C/Y = 12

"The negative indicates that this amount must still be paid back to the bank as the final payment."

Equivalent two-step alternative (2023 Exam 1 Q22 report): N = 299 to find the balance before the final payment ($2605.647…), then N = 1 to add one period's interest ($2614.1157…).

Or the lump adjustment (2018 Exam 1 Q22): run the full N, read the residual FV, and add it to the regular repayment.

Common error the reports name: giving the balance before the last payment instead of balance + interest. "A common incorrect response was $1197.39. This was the balance before the last payment but does not include the interest to be added to the outstanding balance for the last payment." (2021 Q8c)

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2.8 "In total, how much interest …" wording

The method VCAA prescribes (three interchangeable framings, all in reports): 1. Loans: total interest = (number of repayments × repayment) − (reduction in principal)

"Repayments − reduction in the principal = 12 × 800 − (70 000 − 65 076.22) = 9600 − 4923.78 = 4676.22" (2016 Q7a.ii)
"Paid = 78 × $1704.03 = $132 914.34; The principal reduced by: $350 000 − $262 332.33 = $87 667.67; Interest = $132 914.34 − $87 667.67" (2019 Q9b)
"Total repaid = 2228.40 × 5 × 12 = 133 704; Interest paid = Amount repaid − amount borrowed" (2024 Exam 1 Q21) 2. Annuities (payments received): interest = payments received − balance reduction "Balance start Year 2: 480 242.25; Balance end Year 2: 467 131.14; Balance reduction 13 111.11; Payments in Year 2 = 12 × 2800 = 33 600; Interest = 33 600.00 − 13 111.11 = 20 488.89" (2018 Q6b.ii) 3. Investments (no withdrawals): interest = final balance − initial balance "Interest earned = 2545.33 − 2500 = $45.33" (NHT 2019 Q7a)

Named errors: "A common incorrect answer was $4923.78, which is the reduction in the principal over the year. This failed to take into account the $9600 total of repayments made in the year." (2016). "Some students who answered parts a. and b. correctly gave $454.26 as the answer here, which was the total amount Lily was charged rather than the interest." (2017 Q6c).

Consequential marking: "An answer of $20 488.88, based on an unrounded value from Question 6b.i., was also awarded full marks." (2018 Q6b.ii). But: "Some students followed through with the incorrect response they obtained for Question 9a. but did not show the working required for the consequential mark." (2019 Q9b).

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2.9 "The value of the [asset] after n years can be determined using the rule Vn = …"

Template family: - "The amount of money in the account after n years, Vn, can also be determined using a rule. Complete the rule below by writing the appropriate numbers in the boxes provided. Vn = □ × □^n" (2016 Q5d.i) - "A rule of the form Vn = a × b^n can be used to determine the balance of Julie's account after n months. Complete this rule … balance = □ × □^n" (2018 Q4c.i) - "The value of the machine, in dollars, after n years, Vn, could also be determined using a rule of the form Vn = a + bn. Write down this rule for Vn." (2020 Q7d) - "Complete the rule below that gives the value of the coffee machine, Mn, in dollars, after n cups have been produced. Write your answers in the boxes provided. Mn = □ + □ × n" (2021 Q7c) - "Complete the rule below that gives the value of the car, Cn, in dollars, after n kilometres have been travelled. Cn = □ + □ × n" (NHT 2022 Q6b) - "The value of Marlon's guitar after n concerts, Gn, can be determined using a rule. Complete the rule below by writing the appropriate numbers in the boxes provided. Gn = □ − □ × n" (NHT 2019 Q6c) - "ii. Write a rule for Vn in terms of n." (2025 Q8a.ii) - "The value of the car, in dollars, after travelling n kilometres, Wn, can be modelled by a rule. Write the rule for Wn in terms of n." (NHT 2026 Q6c) - MCQ version: "A rule for the balance, Rn, in dollars, after n years is given by" (2020 Exam 1 Q26); "A rule for Gn, the value of the machine after n years is" (2023 Exam 1 Q18); "An equivalent rule to determine the value of the car after n years is" (NHT 2024 Exam 1 Q17).

Examiner statements on the rule/recurrence distinction:

"Some students were unsure of the distinction between a rule and a recurrence relation." — 2020 Q7d
"Many students did not demonstrate understanding of the distinction between a rule and recurrence relation and gave a similar answer again as in part a.i." — 2026-01_2025-GeneralMaths2-report.txt, Q8a.ii (only 50% correct)
"A common error was for students to write the two numbers in the incorrect boxes." — 2018 Q4c.i
"After completing a question, students should read the question again … For example in core Question 7c., students were asked to write a rule for the value of a coffee machine. Since an earlier question had related value in terms of time, many students failed to recognise the change to value in terms of number of cups produced." — 2021 general comments

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2.10 Other high-frequency stem furniture

• "The value of [asset], in dollars, after n years, Vn, can be modelled by the recurrence relation shown below." (2017, 2018, 2019, 2020, 2022, 2023 …)
• "The balance of the loan, in dollars, after n months, Ln, can be modelled by the recurrence relation" (2022 Q8; 2023 NHT Q6; 2026 NHT Q5)
• "Let Vn be the balance of the annuity after n monthly payments." (2021 Q9a)
• "can be determined using a recurrence relation of the form V0 = 60000, Vn+1 = 1.0015Vn − d" (2023 Q7) — invites solving for d.
• "Complete the calculations below by writing the appropriate numbers in the boxes provided." (2017 Q5a)
• "Write your answers in the boxes provided." (2021 Q6c, Q7c; NHT 2022 Q6b)
• "Write your answers in the spaces provided in the table below." (2023 Q6b; NHT 2024 Q7d)
• "For what value of k would this investment act as a simple perpetuity?" (2020 Q10c; Sample Q8c)
• "For what value of d does the recurrence relation generate a geometric sequence?" (2023 Q7d)
• "How much money did [name] initially deposit/invest/borrow?" (2016 Q5a; 2018 Q4a; 2018 NHT Q7a; 2024 NHT Q7a; 2025 Q7a) — the reliable opening 1-mark gift; 87–96% correct every year.
• "What monthly payment does Phil receive?" / "What amount does Timmy repay each month?" (2019 Q8a; 2026 NHT Q5a) — read d straight off the recurrence; still only 72–74% correct.

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3. Traps and examiner complaints (recurring, with years)

3.1 Rounding

3.2 Sign conventions in the finance solver

• 2016 Exam 1 Q24: the flagged weakness of the whole Exam 1 — students had to "correctly apply a sign convention appropriate to an annuity."
• 2018 Exam 1 Q24: "Understanding of the sign convention for the finance solver is very important, as is the careful tracking of values used in subsequent calculations."
• 2019 Exam 2 Q9a: entering FV positive instead of −262 332.33 gave $7954.54; "This very large fortnightly repayment should have been a signal that something had been entered incorrectly."
• 2025 Exam 2 Q10: "Some students found the payment correctly from the calculator but answered as positive 22 216.27" — the recurrence needed + (−22 126.27).
• 2024 NHT Exam 2 Q9a answer V0 = 35 000, Vn+1 = 1.025 × Vn − 1722 — the − matters.

3.3 Confusing a recurrence relation with a rule

• 2016 Q6b: "using a rule for the nth term Cn = 38000 − 2750n instead of the recurrence rule."
• 2017 Q6b: "Some students did not recognise the difference between the recurrence relation above and the rule An = 428 × 1.015^n."
• 2019 Q7d: "A few wrote both a recurrence relation and a rule … which could not be accepted."
• 2020 Q7d: "Some students were unsure of the distinction between a rule and a recurrence relation."
• 2022 Q6c: "A common error was to finish the recurrence relation with 12500n."
• 2024 Q5c / 2024 PL: "It was common to see the n added at the end of an otherwise correct answer."
• 2025 Q8a.ii: "Many students did not demonstrate understanding of the distinction … and gave a similar answer again as in part a.i."

3.4 Not showing the recursion steps

• 2016 Q5b: using the explicit rule Vn = R^n V0 — "not using recursion as required." 27% scored zero.
• 2018 Q4b.i: "some students used methods other than simple recursion, which were not acceptable."
• 2019 Q7a: "To state V1 = 54 000 without showing how this was obtained does not address the instruction to 'show'."
• 2020 Q7b: "Quite a few students did not use recursion as asked. There was evidence of inaccurate copying of numbers with zeros left out." 28% scored zero.
• 2021 Q9a: "Many students correctly calculated the factor of 1.00425 but either added the $900 or did not show the recursive calculations in full." Only 21% got 2/2.
• 2022 Q8a: "Some gave the correct final answer but did not show the recursive calculations as asked. Transcription and rounding errors were common." 56% scored zero.
• 2024 Q7a: "Some students did not show the full detail required in these recursive calculations." 51% scored zero.

3.5 Wrong compounding period / rate conversion

See §2.5 for the quotes. The pattern is symmetrical and repeats every single year: some students give the per-period rate where the annual is wanted, others give the annual where the per-period is wanted. The reports name it in 2017, 2018, 2019, 2020 (twice in one paper), 2021, 2022, 2023, 2024, 2025. Additional named variants: - Ignoring the number of periods in unit-cost depreciation: "$4.40 was found by ignoring the eight years" (2016 Q6c); "Students who divided by 1920 only" (2017 Exam 1 Q21); "$14760 was a common incorrect answer, with some students not accounting for 52 weeks per year" (2024 Q5b). - Confusing years with months: "An answer of 180 years was sometimes seen given by students who may have confused years with months." (2024 PL, Q7b). - Choosing the lowest nominal rate without regard to compounding: 2018 Exam 1 Q19.

3.6 Perpetuity misconceptions

Named in 2015 (legacy), 2018, 2021, 2022, 2024 as a persistent weakness: - "Perpetuities was an area that requires improvement for many students." (2018) - "The definition of a perpetuity continues to be misunderstood by many students." (2021 Q6b) - "Many students were unable to provide the definition of a perpetuity." (2022 Q7d) - "Perpetuities remain an area of concern for students who do not understand that these are a special case of an annuity that lasts indefinitely." (2024 PL) - "Many students did not know that perpetuities pay out only the interest earned, while the principal remains unchanged." (2015 legacy)

3.7 Using formulas instead of technology (and vice versa)

• 2016 Q7a.i: "Rather than use a financial solver to answer the question above, a number of students adopted a formulaic approach, almost always unsuccessfully, based on the compound interest formula."
• 2017 Q7b: "Formulas should not have been used. The finance solver entries required were: …"
• 2018 Q6b.ii: "A few students tried unsuccessfully to use the annuities formula."
• 2015 legacy general comments list of common errors: "using the compound formula for a flat rate question (Module 4: Business-related mathematics, Question 2a.)"
• 2013 legacy Q3a: "Students often confuse simple (flat) interest with compound interest calculations. Many students inappropriately treated this as a reducing balance problem."
• Conversely 2025 Q7c: using a finance solver where the question said "using the values in the table" gave a wrong cents figure — "not appropriate."
• 2021 Q8b: "any response including CAS syntax/notation cannot attract full marks."

3.8 Transcription and presentation

• 2017 Q5a: "A few students made careless transcription errors."
• 2020 Q7b: "inaccurate copying of numbers with zeros left out."
• 2021 general: "Transcription errors can be costly. … It was not uncommon to see answers with a 0 or 1 missing or digits interchanged."
• 2022 general: "An extra zero was often seen in answers to the recursion and financial modelling questions."
• 2024 general: "Transcription errors were often seen, and students should take care copying from one line to the next or from their CAS."
• Every year: "Scanned images are used for assessing … write in a dark colour (for example, blue or black pen or 2B pencil)."

3.9 Over-answering / further engagement

• 2017 general: "Some students correctly answered a question and then added extra irrelevant information. However, this extra information cannot be awarded extra marks and, if it is incorrect, can lead to marks not being awarded."
• 2024 general: "Students need to be careful that if they provide additional information in their response, it must be correct. … where a student further engaged to provide an incorrect delay time in their response, the mark was not awarded."
• 2020 general: "The mark could not be awarded due to this incorrect further engagement."
• 2024 general: "Students are advised not to write sentences when a numerical answer is asked for."

3.10 Not answering the question actually asked

• 2019 Q8a: "The question was straightforward but often misunderstood. Some gave the answer to Question 8d."
• 2019 Q8d: "The answer to Question 8a. was sometimes given."
• 2021 Q7c: "Many students did not recognise that the value was of the machine in terms of the number of cups of coffee made."
• 2022 Q8c: "some students did not clearly indicate the small amount less than one dollar as asked."
• 2024 Q8: "Many students found the 288 but left the total cost blank."
• 2024 general: "Another example of the importance of reading carefully was in Question 8, with 'the final repayment amount being slightly different from all the other repayments'. Some students incorrectly used 289 payments to find a final payment of $4.20 (not slightly different)."
• 2024 general self-check list: "Has the question been fully answered? Does the answer match the value provided in a 'show that' question? Is the answer reasonable in a practical context?"

3.11 Two-mark questions without working

• 2020 general: "For all questions worth more than one mark, students are strongly advised to show their working. An incorrect answer on its own will not be awarded any marks in a two-mark question; however, often a method mark can be awarded … For example, in Core Question 11, a method mark was available for students who correctly found the repayment value of $2400 but were unable to determine the number of further repayments required."
• 2015 legacy: "Organised presentation of a multi-step calculation was essential for high marks. Many students wrote answers without calculations to two-mark questions. Some working may qualify for a method mark or consequential error mark even if the answer is incorrect."
• 2013 legacy Q4: "Students needed to read this question carefully to identify the three steps needed for the solution. Writing and labelling such steps can be very helpful in organising thoughts, but very few students showed any TVM input or other working."
• 2024 marking key: M1 = method mark, A1 = answer mark, H1 = consequential mark. 2024 PL: "A consequential mark will only be given if allowed for on a marking guide, and then only if the working out is shown."

3.12 Multi-stage / changed-condition problems are the reliable discriminator

Every year the report names these as the hardest items: - 2016 Exam 1: Q24 (non-routine annuity). Exam 2: Q7b (8% full marks). - 2017 Exam 1: "Questions 21, 23 and 24" — Q24 rate change after 10 years. Exam 2: Q7b (26%). - 2018 Exam 1: "the comparison of different financial situations (Questions 19 and 20) or more complicated calculations (Questions 22, 23 and 24)"; "problems involving changing conditions were particularly challenging." - 2019 Exam 1: "Students had difficulty with questions that tested a variety of concepts, in particular Questions 20 to 24." Exam 2: Q9b (19%), Q9c (24%). - 2020 Exam 1: "The most challenging questions involved forming a rule for a compound interest investment (Question 26) or a change in condition part way through the problem (Questions 29 and 30)." - 2021 Exam 1: "Students did not answer well the questions involving the use of the finance solver or questions involving a change in condition part way through the problem (Questions 19, 20, 21, 23 and 24)." Exam 2: Q9b (8%). - 2022 Exam 1: "Students did not score as highly on questions involving the use of recurrence relations or the finance solver (Questions 17, 19, 21, 23 and 24)." Exam 2: Q8c (23%), Q8d (19%). - 2023 Exam 1: "Students did not score as highly on questions involving the use of recurrence relations or the finance solver, or where questions required two or more steps." - 2024 Exam 1: "This was particularly the case in questions that required multiple steps or required multiple options being checked, such as … Questions 20, 23 and 24." - 2025 Exam 2: Q10 (11% full marks) — the two-stage annuity-halfway recurrence.

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4. Quick-reference: canonical answer templates

Recurrence relation (1 mark, both parts required):

V0 = 320 000,   Vn+1 = 1.003 Vn − 1600

Rule (1 mark):

Vn = 120 000 − 15 000n            (linear / flat rate / unit cost)
Vn = 15 000 × 1.04^n              (geometric / compound / reducing balance)

Use recursion to show (1 mark):

V0 = 60 000
V1 = 0.9 × 60 000 = 54 000
V2 = 0.9 × 54 000 = 48 600

Show that a rate is r% (1 mark):

r/1200 = 0.0035
r = 1200 × 0.0035 = 4.2%

or

(1.001 − 1) × 26 × 100 = 2.6%

Finance solver block (loan):

N     = 60
I%    = 3.72
PV    = 175 260.56
PMT   = −3200
FV    = −368.12
P/Y = C/Y = 12

Finance solver block (investment / annuity):

N     = 96
I%    = 3.20
PV    = −307 794.50
PMT   = −1854.05
FV    = 600 000
P/Y = C/Y = 12

Total interest (loan): interest = (n × repayment) − (principal reduction) Total interest (annuity): interest = (n × payment received) − (balance reduction) Final repayment: N = 1, PV = balance after second-last payment, FV = 0, solve PMT Amortisation next line: interest = rate × previous balance; PR = payment − interest; balance = previous − PR