Annuity investments and savings plans: VCE General Mathematics Unit 3 Recursion and financial modelling

Syllabus dot point

“Compound interest investment with periodic and equal additions to the principal: use of a first-order linear recurrence relation to model and analyse (numerically and graphically) annuity investment, use of a table to analyse its growth step by step, and use of technology with financial modelling functionality to find the future value, the number of compounding periods for the investment to exceed a given value, and the interest rate or payment amount needed for an investment to exceed a given value in a given time”

13 min readUpdated 23 Sep 2026

Quick answer

An annuity investment (savings plan) earns compound interest and has an equal deposit added every period: $V_0 =$ initial amount, $V_{n+1} = RV_n + d$ , where $R = 1 + \frac{r}{100}$ for a rate of $r\%$ per compounding period. Each period, interest $=$ balance $\times \frac{r}{100}$ and the new balance $=$ old balance $+$ interest $+$ deposit. On the finance solver, money paid in ( $PV$ and $PMT$ ) is negative and the final balance ( $FV$ ) is positive; solve for whichever of $N$ , $I\%$ , $PMT$ or $FV$ is unknown. Total interest $=$ final balance $-$ initial amount $-$ total deposits.

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What this dot point is asking
The answer
Investments, loans and annuities: one model
Exam-style questions
Practice questions

What this dot point is asking

The Study Design topic "Compound interest investment with periodic and equal additions to the principal" lists:

"use of a first-order linear recurrence relation to model and analyse (numerically and graphically) annuity investment, including the use of a recurrence relation to determine the value of the investment after $n$ payments have been made for an initial sequence from first principles";
"use of a table to investigate and analyse the growth of an annuity investment on a step-by-step basis after each payment is made, the payment made, the interest earned and the balance of the investment";
"use of technology with financial modelling functionality to solve problems involving annuity investments, including determining the future value of an investment after a number of compounding periods, the number of compounding periods for the investment to exceed a given value and the interest rate or payment amount needed for an investment to exceed a given value in a given time".

So you need to write and iterate the recurrence, complete a growth table, interpret the balance graph, and set up the finance solver to find FV, N, I% or PMT. In 2025 Examination 1 Question 22, $68\%$ of students found the interest rate needed for a savings account to reach $14 000.

The answer

The recurrence relation

Let $V_n$ be the balance after $n$ deposits. If interest is $r\%$ per compounding period and a deposit $d$ is made after the interest is credited each period:

V_0 = \text{initial amount}, \qquad V_{n+1} = RV_n + d, \qquad R = 1 + \frac{r}{100}.

For a nominal annual rate compounded $k$ times a year, $r = \frac{\text{annual rate}}{k}$ . For $3.6\%$ per annum compounding monthly, $r = 0.3\%$ and $R = 1.003$ .

This is the same first-order linear recurrence as a reducing balance loan or an annuity, with the sign of $d$ telling you the direction: $+d$ means money is added (an investment grows), $-d$ means money is removed (a loan or annuity balance falls). With $d = 0$ it is plain compound interest, a geometric sequence.

The growth table

A table breaks each period into its parts, exactly like an amortisation table for a loan:

Payment number	Payment (dollars)	Interest (dollars)	Balance (dollars)
0	0.00	0.00	4800.00
1	200.00	14.40	5014.40
2	200.00	15.04	5229.44
3	200.00	15.69	5445.13

This plan starts with $4800, earns $3.6\%$ per annum compounding monthly ( $0.3\%$ per month), and adds $200 each month. For each row:

interest $=$ previous balance $\times 0.003$ (for row 2, $5014.40 \times 0.003 = 15.0432 \approx 15.04$ );
balance $=$ previous balance $+$ interest $+$ payment ( $5014.40 + 15.04 + 200 = 5229.44$ ).

The interest grows every row because it is earned on a larger balance each time. In the exam, use the values shown in the table for the next row, as the 2025 Examination 2 amortisation question required.

The balance graph

Plotted against time, the balance rises along a curve that gets steeper: the deposits alone would give a straight line, and interest earned on interest bends the curve upwards. The vertical gap between the two is the total interest earned so far.

The finance solver

TI-Nspire (Finance Solver) and ClassPad (Financial, Compound Interest) use the same seven fields:

Field	Meaning	Savings plan entry
$N$	number of deposits (compounding periods)	years $\times$ periods per year
$I\%$	annual interest rate	e.g. 3.6
$PV$	starting balance	negative (paid in)
$PMT$	regular deposit	negative (paid in)
$FV$	balance at the end	positive (you can take it out)
$P/Y$ , $C/Y$	payments and compounds per year	both 12 for monthly

Fill in six fields and solve for the seventh. The sign rule is cash flow: money leaving your pocket is negative, money coming back is positive. If $PV$ and $PMT$ have different signs, the solver models a loan or annuity being drawn down, not a savings plan.

The four standard questions:

Future value: solve for $FV$ .
Number of periods to exceed a value: enter the target as $FV$ and solve for $N$ , then round up to the next whole period (a part period is not a real deposit, and rounding down leaves you short).
Interest rate needed: solve for $I\%$ (2025 Examination 1 Question 22).
Payment needed: solve for $PMT$ and round up to the next cent so the target is actually reached.

Interest earned

Your own money is the initial amount plus every deposit, so

\text{total interest} = \text{final balance} - \text{initial amount} - (\text{number of deposits} \times d).

The interest earned in a particular year is the change in balance over that year minus the deposits made in it.

Formula

$V_0 =$ initial amount, $V_{n+1} = RV_n + d$ with $R = 1 + \frac{r}{100}$ ( $r\%$ per period). Interest for a period $=$ balance $\times \frac{r}{100}$ . New balance $=$ balance $+$ interest $+ d$ . Finance solver: $PV < 0$ , $PMT < 0$ , $FV > 0$ for a savings plan. Total interest $=$ FV $-$ PV $-$ total deposits (using the sizes, not the signs).

Worked examples: recurrence, solver and interest

The recurrence from first principles

The plan in the table: $4800 at $3.6\%$ per annum compounding monthly with $200 added each month.

Step 1: the growth factor: $r = \frac{3.6}{12} = 0.3\%$ per month, so $R = 1.003$ .
Step 2: the recurrence: $V_0 = 4800$ , $V_{n+1} = 1.003V_n + 200$ .
Step 3: iterate: $V_1 = 1.003 \times 4800 + 200 = 5014.40$ . $V_2 = 1.003 \times 5014.40 + 200 = 5229.4432$ . $V_3 = 1.003 \times 5229.4432 + 200 = 5445.13$ (to the nearest cent). Keep the unrounded value in your calculator between steps; round only the answer.

Future value after five years

Entries: $N = 60$ , $I\% = 3.6$ , $PV = -4800$ , $PMT = -200$ , $FV =$ solve, $P/Y = C/Y = 12$ .
Result: $FV = 18\,871.42$ . The balance after five years is $18 871.42.
Interest: own money $= 4800 + 60 \times 200 = 16\,800$ dollars, so interest $= 18\,871.42 - 16\,800 = 2071.42$ dollars.

How long to exceed $20 000?

Entries: $N =$ solve, $I\% = 3.6$ , $PV = -4800$ , $PMT = -200$ , $FV = 20\,000$ , $P/Y = C/Y = 12$ .

Result: $N = 64.38$ . After 64 months the balance is $19 902.50 (still short); after 65 it is $20 162.21. Answer: 65 months.

The interest rate needed (2025 Examination 1 Question 22)

Entries: $N = 156$ , $I\% =$ solve, $PV = -4000$ , $PMT = -50$ , $FV = 14\,000$ , $P/Y = C/Y = 52$ .
Result: $I\% = 8.37\ldots$ , so the rate is about $8.4\%$ per annum. Answer: A.
Recurrence form: $R = 1 + \frac{8.3723}{5200} = 1.00161$ , so $V_0 = 4000$ , $V_{n+1} = 1.00161V_n + 50$ .

The deposit needed

A saver has $10 000 and wants $50 000 in eight years, with interest at $4.2\%$ per annum compounding monthly. What monthly deposit is needed?

Entries: $N = 96$ , $I\% = 4.2$ , $PV = -10\,000$ , $PMT =$ solve, $FV = 50\,000$ , $P/Y = C/Y = 12$ .

Result: $PMT = -316.301\ldots$ A deposit of $316.30 falls short by about 14 cents after 96 months, so the saver must deposit $316.31 a month.

Interest earned in the second year

For the $4800 plan, the balance after 12 months is $7415.68 and after 24 months is $10 127.09 (solve for $FV$ with $N = 12$ and $N = 24$ ). During year 2 the saver deposited $12 \times 200 = 2400$ dollars, so

\text{interest in year 2} = 10\,127.09 - 7415.68 - 2400 = 311.41 \text{ dollars}.

A rate change part-way through

After three years at $3.6\%$ the $4800 plan holds $12 937.74 ( $N = 36$ ). The rate then rises to $4.8\%$ per annum for the last two years, with the same deposits.

Stage 2 entries: $N = 24$ , $I\% = 4.8$ , $PV = -12\,937.74$ , $PMT = -200$ , $P/Y = C/Y = 12$ : $FV = 19\,266.02$ .

The balance after five years is $19 266.02, which is $394.60 more than the $18 871.42 it would have been at $3.6\%$ throughout. A change of rate, deposit or timing always means a new solver calculation starting from the balance at the time of the change.

Common traps

Wrong signs on the finance solver: For a savings plan $PV$ and $PMT$ are both negative and $FV$ is positive. A positive $PMT$ with a negative $PV$ is a withdrawal, which models an annuity paying you, not a savings plan.
Using the annual rate as the period rate: In the recurrence, $R = 1 + \frac{3.6}{1200} = 1.003$ for monthly compounding, not $1.036$ . On the solver, $I\%$ is the annual rate and $P/Y$ , $C/Y$ do the dividing.
$N$ in years: $N$ counts periods: three years of weekly deposits is $N = 156$ .
Rounding $N$ or $PMT$ down: For "first exceeds" or "enough to reach", round up.
Counting deposits as interest: Subtract the initial amount and every deposit.
Rounding between recursive steps: Keep full accuracy in the calculator; VCAA reports note that early rounding loses the mark.
Mixing up the order: VCE models add the interest and then the deposit, $V_{n+1} = RV_n + d$ . Read the question: "after the interest is added, a deposit is made".

Exam technique

Write the seven solver entries in your working as a list ( $N = 156$ , $I\% = ?$ , $PV = -4000$ , ...). Markers can award method marks from a correct set of entries even if the final answer is mistyped, and writing them out is the quickest way to spot a sign or period error. For multiple choice, estimate first: total your own deposits, and any option that implies a large amount of interest from a low rate (or the reverse) can be eliminated.

Investments, loans and annuities: one model

The whole Recursion and financial modelling topic runs on one recurrence, $V_{n+1} = RV_n + d$ :

Situation	$R$	$d$	Balance
Compound interest investment	$R > 1$	$0$	grows geometrically
Annuity investment (savings plan)	$R > 1$	$d > 0$ (deposit)	grows faster than compound interest
Reducing balance loan	$R > 1$	$d < 0$ (repayment)	falls to zero
Annuity (regular income)	$R > 1$	$d < 0$ (withdrawal)	falls to zero
Perpetuity	$R > 1$	$-d =$ interest each period	stays constant

On the finance solver the same pattern appears in the signs. A savings plan has $PV$ and $PMT$ with the same sign (both paid in); a loan or an annuity drawdown has $PV$ and $PMT$ with opposite signs. Many exam questions chain two of these: saving up with an annuity investment, then drawing the balance down as an annuity.

Note

A savings plan is an account that earns interest and that you also keep adding to. Each period, the bank adds interest on what is already there, then you add your regular deposit. On the calculator's finance solver, the money you put in (the starting amount and each deposit) is negative and the amount you end up with is positive. The interest you earned is what you end up with minus everything you put in.

Exam-style questions

Questions in the style of VCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

2025 VCAA Exam 1 Q22Past paper1 mark

Rita opens a savings account with an initial deposit of $4000. The account earns interest compounding weekly. 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 $14 000 after three years is closest to A. 8.4% B. 14.2% C. 14.6% D. 17.2%

Show worked answer →

A starting balance plus a regular deposit after the interest each week is an annuity investment. Finance solver entries:

$N$	$I\%$	$PV$	$PMT$	$FV$	$P/Y$	$C/Y$
$3 \times 52 = 156$	solve	$-4000$	$-50$	$14\,000$	52	52

$PV$ and $PMT$ are negative because that money is paid into the account; $FV$ is positive because it can be taken out at the end. The solver gives $I\% = 8.3723\ldots$ , so the rate is about 8.4%, option A ( $68\%$ correct).

Check: at $8.4\%$ the balance after 156 weeks is about $14 008. Rita deposits $4000 + 156 \times 50 = 11\,800$ dollars of her own money, so only about $2200 of the $14 000 is interest; a rate near $14\%$ to $17\%$ would give far more than $14 000.

Source: VCAA 2025 General Mathematics Examination 1, Question 22, and the 2025 examination report.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation2 marks

A savings plan is modelled by

V_0 = 3000

V_{n+1} = 1.0025V_n + 100

, where

V_n

is the balance in dollars after

n

months. (a) State the initial investment and the monthly deposit. (b) Find the annual interest rate.

Show worked solution →

(a) Initial investment $3000; monthly deposit $100 (added, so the balance grows). (1 mark)

(b) Monthly rate $= 1.0025 - 1 = 0.0025 = 0.25\%$ , so the annual rate is $0.25\% \times 12 = 3\%$ per annum, compounding monthly. (1 mark)

foundation3 marks

A savings plan starts with $2000, and $150 is added at the end of each month after interest of

4.8\%

per annum, compounding monthly, is credited. (a) Write a recurrence relation for the balance

V_n

after

n

months. (b) Showing recursive calculations, find the balance after three months, to the nearest cent. (c) Find the total interest earned in those three months.

Show worked solution →

(a) Monthly rate $\frac{4.8}{12} = 0.4\%$ , so $R = 1.004$ : $V_0 = 2000$ , $V_{n+1} = 1.004V_n + 150$ . (1 mark)

(b) $V_1 = 1.004 \times 2000 + 150 = 2158$ ; $V_2 = 1.004 \times 2158 + 150 = 2316.632$ ; $V_3 = 1.004 \times 2316.632 + 150 = 2475.898\ldots$ The balance is $2475.90. (1 mark)

(c) Interest $= 2475.90 - 2000 - 3 \times 150 = 25.90$ dollars. (1 mark)

core2 marks

A row of a savings plan's table shows a balance of $12 000 after payment 5. Interest is

0.35\%

per month and $400 is deposited each month after the interest. Complete the rows for payments 6 and 7 (interest and balance), to the nearest cent.

Show worked solution →

Payment 6: interest $= 0.0035 \times 12\,000 = 42.00$ ; balance $= 12\,000 + 42.00 + 400 = 12\,442.00$ . (1 mark)

Payment 7: interest $= 0.0035 \times 12\,442 = 43.547 \approx 43.55$ ; balance $= 12\,442.00 + 43.55 + 400 = 12\,885.55$ . (1 mark)

core2 marks

An investor deposits $500 at the end of every quarter into an account that starts at $0 and earns

5.2\%

per annum, compounding quarterly. (a) Find the balance after 10 years, to the nearest cent. (b) How much of it is interest?

Show worked solution →

(a) $N = 40$ , $I\% = 5.2$ , $PV = 0$ , $PMT = -500$ , $FV =$ solve, $P/Y = C/Y = 4$ : $FV = 26\,015.41$ . The balance is $26 015.41. (1 mark)

(b) Deposits total $40 \times 500 = 20\,000$ dollars, so the interest is $26\,015.41 - 20\,000 = 6015.41$ dollars. (1 mark)

core2 marks

A savings account starts with $1000 and $100 is added every fortnight. Interest is

4.16\%

per annum, compounding fortnightly (26 fortnights a year). After how many fortnights will the balance first exceed $10 000?

Show worked solution →

Finance solver: $N =$ solve, $I\% = 4.16$ , $PV = -1000$ , $PMT = -100$ , $FV = 10\,000$ , $P/Y = C/Y = 26$ : $N = 82.9\ldots$ (1 mark)

After 82 fortnights the balance is about $9894.84, still short; after 83 it is about $10 010.67. So the balance first exceeds $10 000 after 83 fortnights. Always round $N$ up for "first exceeds". (1 mark)

exam2 marks

A family has $20 000 in a savings account and plans to add $250 at the end of each month. What annual interest rate, compounding monthly, is needed for the balance to reach $45 000 after six years? Give the answer as a percentage correct to two decimal places, and show the finance solver entries.

Show worked solution →

$N = 72$ , $I\% =$ solve, $PV = -20\,000$ , $PMT = -250$ , $FV = 45\,000$ , $P/Y = C/Y = 12$ . (1 mark)

$I\% = 3.6607\ldots$ , so the rate needed is 3.66% per annum. (1 mark) (Deposits alone total $20\,000 + 72 \times 250 = 38\,000$ dollars, so about $7000 must come from interest.)

exam3 marks

An account starts with $15 000 and $300 is added at the end of each month for four years, at

4.5\%

per annum compounding monthly. The deposits then stop, and the balance is left to earn interest at the same rate for three more years. (a) Find the balance after four years. (b) Find the balance after seven years. (c) Find the total interest earned over the seven years. Give answers to the nearest cent.

Show worked solution →

(a) $N = 48$ , $I\% = 4.5$ , $PV = -15\,000$ , $PMT = -300$ , $P/Y = C/Y = 12$ : $FV = 33\,697.37$ dollars. (1 mark)

(b) New entries: $N = 36$ , $PV = -33\,697.37$ , $PMT = 0$ : $FV = 38\,558.14$ dollars. With no deposits this stage is ordinary compound interest. (1 mark)

(c) Own money $= 15\,000 + 48 \times 300 = 29\,400$ dollars, so interest $= 38\,558.14 - 29\,400 = 9158.14$ dollars. (1 mark)

Annuity investments and savings plans: VCE General Mathematics Unit 3 Recursion and financial modelling

What this dot point is asking

The answer

The recurrence relation

The growth table

The balance graph

The finance solver

Interest earned

The recurrence from first principles

Future value after five years

How long to exceed $20 000?

The interest rate needed (2025 Examination 1 Question 22)

The deposit needed

Interest earned in the second year

A rate change part-way through

Investments, loans and annuities: one model

Exam-style questions

Practice questions

Practise this

Sources & how we know this