SCSA WACE 2021-style practice: \8000is invested at4.8\%per annum compounding monthly. (a) Write a recurrence relation for the balanceA_nafternmonths. (b) Find the balance after2$ years. (c) Find the effective annual interest rate, correct to two decimal places.

Compounding monthly means the monthly rate is the nominal annual rate divided by 12. (a) Monthly rate = 4.8\%12 = 0.4\% = 0.004. Recurrence: A_n+1 = 1.004\,A_n, with A_0 = 8000. (2 marks) (b) Two years is 24 months, so A_24 = 8000 × 1.004^24 = 8000 × 1.10057 = 8804.55, that is \8804.55$. (2 marks) (c) Effective annual rate = 1.004^12 - 1 = 1.04907 - 1 = 0.04907 = 4.91\%. (2 marks) Markers reward the per-period rate, the correct number of periods, and an effective rate from one year of compounding.

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WAMathematics ApplicationsSyllabus dot point

How does recursion model a compound interest investment that grows each period?

Model compound interest with a recurrence relation, convert nominal annual rates to the period rate, and find balances and effective rates.

How to model a compound interest investment with a recurrence relation, convert a nominal annual rate to the compounding-period rate, find any balance, and compare effective annual rates.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The compound interest recurrence
Converting nominal rates
The effective annual rate
Using the recurrence on a calculator

What this dot point is asking

You must set up the recurrence, convert nominal rates to the per-period rate, find any balance, and calculate and compare effective annual rates.

The compound interest recurrence

Each period the balance is multiplied by a fixed growth factor, so it is a geometric model.

The key word is period. If interest compounds monthly, $n$ counts months and $i$ is the monthly rate; if it compounds quarterly, $n$ counts quarters. Mixing years with a monthly rate is the most common error in the topic.

Converting nominal rates

A nominal annual rate is the headline yearly rate. The rate actually applied each period is the nominal rate divided by the number of compounding periods per year.

Compounding monthly: $i = \dfrac{\text{nominal}}{12}$ , and $n$ counts months.
Compounding quarterly: $i = \dfrac{\text{nominal}}{4}$ , and $n$ counts quarters.
Compounding daily: $i = \dfrac{\text{nominal}}{365}$ , and $n$ counts days.

For a nominal $7.2\%$ per annum compounding monthly, $i = \dfrac{0.072}{12} = 0.006$ , so $R = 1.006$ .

Worked example

Finding a balance and the interest earned

$\$12\,000$ is invested at $6\%$ per annum compounding quarterly for $3$ years. Find the balance and the total interest.

Step 1: Convert the nominal rate to the period rate

The nominal annual rate must be divided by the number of compounding periods per year to get the rate that actually applies each period. Quarterly rate $i = \dfrac{0.06}{4} = 0.015$ , so the growth factor $R = 1.015$ .

Step 2: Count the total number of compounding periods

Because interest is applied quarterly, time must be counted in quarters rather than years. Three years compounding quarterly gives $n = 3 \times 4 = 12$ periods.

Step 3: Apply the explicit rule to find the balance

Raising the growth factor to the number of periods gives the balance directly from the principal. $A_{12} = 12000 \times 1.015^{12} = 12000 \times 1.19562 = 14347.45$ , so the balance is $\$14\,347.45$ .

Step 4: Find the total interest

The interest earned is the final balance minus the original principal. Interest $= 14347.45 - 12000 = 2347.45$ , so $\$2347.45$ in total. Notice the interest is more than $6\% \times 3 = 18\%$ of the principal because each quarter earns interest on the interest already added.

Final answer: Balance after $3$ years is $\$14\,347.45$ ; total interest earned is $\$2347.45$ .

The effective annual rate

Two accounts with the same nominal rate are not equally good if they compound at different frequencies. The effective annual rate restates a nominal rate as the equivalent rate compounding once a year, so accounts can be compared fairly.

For a nominal $6\%$ compounding monthly, $i = 0.005$ and $i_{\text{eff}} = 1.005^{12} - 1 = 0.06168 = 6.17\%$ . The more frequent the compounding, the higher the effective rate, because interest is earned on interest sooner.

Using the recurrence on a calculator

To generate balances term by term, store the principal, then repeatedly multiply by $R$ . This is how the SCSA calculator-assumed section expects you to verify a balance or find when an investment first passes a target. To find the first $n$ for which $A_n$ exceeds, say, $\$20\,000$ , increase $n$ until the balance crosses the threshold, then report the first whole period above it.

Exam-style practice questions

Practice questions written in the style of SCSA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

WACE 20216 marks

\

8000

is invested at

4.8\%

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

A_n

after

months. (b) Find the balance after

2$ years. (c) Find the effective annual interest rate, correct to two decimal places.

Show worked answer →

Compounding monthly means the monthly rate is the nominal annual rate divided by $12$ .

(a) Monthly rate $= \dfrac{4.8\%}{12} = 0.4\% = 0.004$ . Recurrence: $A_{n+1} = 1.004\,A_n$ , with $A_0 = 8000$ . (2 marks)

(b) Two years is $24$ months, so $A_{24} = 8000 \times 1.004^{24} = 8000 \times 1.10057 = 8804.55$ , that is $\$8804.55$ . (2 marks)

Markers reward the per-period rate, the correct number of periods, and an effective rate from one year of compounding.

WACE 20234 marksTwo banks offer the same nominal rate of

6\%

per annum. Bank A compounds annually and Bank B compounds quarterly. Show that Bank B gives the higher effective annual rate and state the difference correct to two decimal places.

Show worked answer →

Compare the growth factor over one year for each bank.

Bank A effective rate $= 6.00\%$ because it compounds once a year. Bank B quarterly rate $= \dfrac{6\%}{4} = 1.5\% = 0.015$ , so its effective rate $= 1.015^4 - 1 = 1.06136 - 1 = 0.06136 = 6.14\%$ . (3 marks)

Bank B is higher by $6.14\% - 6.00\% = 0.14\%$ , because more frequent compounding earns interest on interest sooner. (1 mark)

Markers reward the quarterly rate, the one-year growth factor raised to the fourth power, and the correct difference.