Question 17
An investment of $50 000 that compounds interest monthly is modelled by the recurrence relation $A_{n+1}=1.00375A_n$ where $A_0=50\,000$.
What would be the advertised interest rate per annum, compounding monthly?
[2 marks]How many months would it take for the value of the investment to exceed $51 000?
[2 marks]The multiplier hides a monthly rate, and month 5 falls just short
First the multiplier is pulled apart to find the rate. Then the balance grows one month at a time. The vertical axis starts at 49 800 so the monthly steps are visible.
The multiplier gives you the monthly rate, and twelve of them make the advertised yearly rate. The balance needs six multiplications to get past 51 000.
Name the yearly rate, then run the recurrence past 51 000
For part a), pick the rate you would write as the advertised rate. For part b), press the button once per month, the way you would repeat Ans × 1.00375 on your calculator, then decide how many months it takes.
One step of the recurrence is one month. Work out what the 0.00375 in the multiplier means first, then decide what the question calls the rate “per annum”.
0.375% is the rate for one month, because the relation multiplies once a month. The question asks for the rate per annum, so there is one more step: $12\times0.375\%=4.5\%$.
The decimal point has slipped. To turn 0.00375 into a percentage, multiply by 100: that gives 0.375%, a monthly rate. Then multiply by 12 for the yearly rate.
That is the effective annual rate, . It is real, but the question asks for the advertised rate per annum, compounding monthly. That is the nominal rate, 12 times the monthly rate.
That’s it. The interest rate is 0.375% per month, and the advertised rate is 12 times that: 4.5% p.a. compounding monthly.
Press the button to apply the recurrence once. Keep going until a balance goes past 51 000, then choose your answer.
After 5 months the balance is $A_5=50\,944.56$, still below 51 000. “Exceed” means go past, so you need the first month whose balance is above 51 000.
One month too many. $A_6=51\,135.60$ is already above 51 000, so month 7 is not the first month that exceeds it.
6 months is right. Now show it: keep pressing until your list reaches the first balance above 51 000. The responses the subject report printed listed the amounts leading up to and past 51 000.
That’s it. $A_5=50\,944.56$ is below 51 000 and $A_6=51\,135.60$ is above, so it takes 6 months. Writing both values shows exactly where the balance crosses.
Work through the solution one mark at a time
Try each step yourself before you reveal it. Each part has a mark for the method and a mark for the answer.
The formula book’s recurrence relation for compound interest is $A_{n+1}=rA_n$ with $r=1+i$, where $i$ is the interest rate per compounding period. Here one period is one month. Substitute the multiplier.
$$\begin{aligned}1.00375&=1+i\\i&=0.00375\ \text{per month}\end{aligned}$$
The 2022 marking guide wrote the rule as $r=1+\frac{i}{n}$, with $i$ the yearly rate and $n=12$, giving $1.00375=1+\frac{i}{12}$. The guide’s $\frac{i}{12}$ is the 2025 formula book’s $i$, so both routes are the same working, and the guide accepts alternative approaches.
Marker: correctly substitutes into an appropriate rule
The advertised rate per annum is the monthly rate times the twelve months in a year.
$$12\times0.00375=0.045$$
The annual interest rate is 4.5% p.a. compounding monthly. Equivalent forms such as 0.045 or $\frac{45}{1000}$ are accepted, and follow-through applies if your substitution had a slip. Don’t give 0.375%, which is the monthly rate, or the effective annual rate, which the question did not ask for.
Marker: calculates annual interest rate
Run the recurrence and write each balance until one passes 51 000. On your calculator, enter 50 000, then type Ans × 1.00375 and press equals once per month.
The compound interest rule works too. Write $51\,000=50\,000\times1.00375^n$ and try values: $n=5$ gives $50 944.56 and $n=6$ gives $51 135.60. Logarithms are also accepted: $n>\frac{\ln1.02}{\ln1.00375}\approx5.29$, so the first whole month is 6. Small rounding differences in the balances are accepted.
Marker: correctly uses an appropriate method
Month 5 is still short, so the answer is the next month.
$$\begin{aligned}A_5&=50\,944.56<51\,000\\A_6&=51\,135.60>51\,000\end{aligned}$$
Follow-through applies, so if a slip earlier changed your balances, the first month your own list passes 51 000 still earns this mark. The guide also lists “after 5 months” as an accepted equivalent response, but 6 months is the clearest answer to “how many months”.
Marker: determines when the investment would exceed $51 000
Each phrase of the question gave you something
The recurrence relation is handed to you, so the work is reading it and running it. The 2022 subject report printed responses to both parts as effective practice: substitute into an appropriate rule to get the annual rate, then use recursion or the compound interest rule with the working shown. The 2025 formula book writes $i$ as the rate per period, where the 2022 guide used $\frac{i}{n}$, but the skill and the answer have not changed.
If I had to guess where you might lose marks, it would be stopping one step early in each part. In part a), 0.375% is the monthly rate, and the question wants the yearly one. In part b), $50 944.56 looks close enough after 5 months, but it has not exceeded $51 000 yet. List the balances until one goes past, and the answer of 6 months is sitting in front of you.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2022 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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