Simple interest and AP

Simple interest problems provide a natural context for arithmetic progressions, a core topic in the HSC Sequences and Series syllabus. By recognising that the same dollar amount of interest accrues each year, you gain a reliable shortcut: the balance after (n) years follows an arithmetic sequence whose first term is the principal and whose common difference is the fixed yearly interest. Working through this problem will strengthen your ability to translate real-world financial language into mathematical formulas, and to solve inequalities that determine the time needed to reach a savings target.

Problem Statement

A principal of $2,000,000 is invested at $5.7\%$ p.a. simple interest. Let $A_n$ be the total amount after $n$ years.

• Find $A_1,A_2,A_3,A_4$.
• Find a formula for $A_n$ and evaluate $A_{12}$.
• How many whole years before the amount exceeds $6,000,000?

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Hints

Simple interest adds a fixed amount each year, so $(A_n)$ is arithmetic.

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Solutions

Because the interest is simple, the yearly addition is constant. We first calculate that annual interest amount by multiplying the principal by the rate.

Yearly interest is $0.057\times 2{,}000{,}000=114{,}000$.

Since the balance grows by the same $114,000 each year, $A_n$ is an arithmetic progression with first term $2{,}000{,}000$ and common difference $114{,}000$. We can immediately write the general term.

Hence

$$A_n=2{,}000{,}000+114{,}000n.$$

Substituting the first few values of $n$ gives the amounts after one, two, three, and four years.

So

$$A_1=2{,}114{,}000,\ A_2=2{,}228{,}000,\ A_3=2{,}342{,}000,\ A_4=2{,}456{,}000,$$

For $n=12$, we simply replace $n$ with $12$ in the formula.

and

$$A_{12}=3{,}368{,}000.$$

To find when the balance first passes $6,000,000, we set up an inequality using the same arithmetic formula and solve for $n$. Because the interest earns only once per year, we must round up to the next whole year.

For exceeding $6,000,000:

$$2{,}000{,}000+114{,}000n>6{,}000{,}000 \implies n>35.087\ldots$$

so $n=36$ years.

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Takeaways

• Simple interest generates an arithmetic progression: $A_n = P + n \cdot (P \cdot r)$.
• When solving “exceeds a target” problems with whole-year compounding, always round the inequality solution up to the next integer.
• Mapping a financial scenario onto a known sequence type lets you use all the tools of that sequence – explicit formulas, inequalities, and graphing.

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Further Readings

HSC Complex Numbers, HSC Vectors, HSC Sequences, HSC Last Resorts