Model a compound-interest investment or loan with a recurrence relation practice questions | Year 10 Mathematics

Compound-interest recurrence

Compound interest multiplies the balance by a fixed factor at every period. Each new balance becomes the starting balance for the next period, so the interest in later periods can increase

V_0=P,\quad V_{n+1}=RV_n

Interest on the current balance In each period, simple interest is calculated on the balance at that period's start, then retained $\displaystyle V_{n+1}=V_n+\frac{r}{100}V_n$
Factor the repeated operation The fixed multiplier includes the balance already present $\displaystyle V_{n+1}=\left(1+\frac{r}{100}\right)V_n$
Count completed periods V₀ is the initial principal. Applying the multiplier n times gives the direct formula $\displaystyle V_n=P\left(1+\frac{r}{100}\right)^n$
Separate the interest asked for One-period interest uses consecutive exact balances. Total interest compares the final balance with the principal $\displaystyle \begin{gathered}I_n=V_n-V_{n-1}\\I_{\rm total}=V_n-P\end{gathered}$
State the model assumptions Keep the rate fixed, retain interest, and assume no fees, repayments, deposits or withdrawals. Use full precision internally and round the final reported amounts

Worked example

A simplified loan model starts at $5,000, with 6% annual interest, no repayments and no fees for 3 years. Find the amount owed after 3 years and the interest charged in year 3.

\begin{gathered}V_0=\$5{,}000\\r=6\%\\V_{n+1}=RV_n\end{gathered}

$\displaystyle \begin{gathered}R=1+\frac{6}{100}\\R=1.06\end{gathered}$Form the annual multiplier
$\displaystyle \begin{gathered}V_1=1.06\times\$5{,}000\\V_1=\$5{,}300\end{gathered}$Apply the multiplier once
$\displaystyle \begin{gathered}V_2=1.06\times\$5{,}300\\V_2=\$5{,}618\end{gathered}$Multiply the previous balance by 1.06 to find the balance after year 2.
$\displaystyle \begin{gathered}V_3=1.06\times\$5{,}618\\V_3=\$5{,}955.08\end{gathered}$Apply the same multiplier to the year-2 balance for year 3.
$\displaystyle \begin{gathered}I_3=V_3-V_2\\I_3=\$5{,}955.08-\$5{,}618\\I_3=\$337.08\end{gathered}$Subtract the balance at the start of year 3 from the balance at its end

Common mistakes

Using simple interest for compound growth

✕$\displaystyle \begin{gathered}V_2=\$2{,}000+2\times\$100\\V_2=\$2{,}200\end{gathered}$✓$\displaystyle \begin{gathered}V_2=\$2{,}000\left(1+\frac{5}{100}\right)^2\\V_2=\$2{,}205\end{gathered}$

The second year's interest uses the updated balance, so it includes interest on the first year's interest

Using the rate as the multiplier

✕$\displaystyle \begin{gathered}V_2=\$2{,}000\left(\frac{5}{100}\right)^2\\V_2=\$5\end{gathered}$✓$\displaystyle \begin{gathered}V_2=\$2{,}000\left(1+\frac{5}{100}\right)^2\\V_2=\$2{,}205\end{gathered}$

The multiplier is 1 plus the rate as a decimal because the balance remains as well as gaining interest

Calling cumulative interest one-period interest

✕$\displaystyle \begin{gathered}I_3=V_3-V_0\\I_3=\$5{,}955.08-\$5{,}000\\I_3=\$955.08\end{gathered}$✓$\displaystyle \begin{gathered}I_3=V_3-V_2\\I_3=\$5{,}955.08-\$5{,}618\\I_3=\$337.08\end{gathered}$

Interest in year 3 is the change during year 3, whereas cumulative interest compares the final balance with the starting balance

Difficulty levels

introUse a supplied recurrence relation with a whole rate to find three exact balances

standardDerive a multiplier from a one-decimal rate, find balances to the nearest cent and calculate total interest

challengeModel a growing loan, find interest in a named period and compare the result with simple interest

examChoose a balance from options based on simple growth, multiplier and iteration errors

Questions about this topic

Why is the previous balance used in compound interest?+

Interest is added to the balance at the end of each period. The next period therefore starts with the updated balance

How is interest in one period different from total interest?+

Interest in one period is the difference between two consecutive balances. Total interest is the final balance minus the starting balance

What does the recurrence relation mean?+

It means the next balance is found by multiplying the current balance by the fixed multiplier R. The index n counts completed compounding periods

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 3: Algebra and indices

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Model a compound-interest investment or loan with a recurrence relation

Repeated simple interest produces compound interest

Compound-interest recurrence

Worked example

Common mistakes

Difficulty levels

Questions about this topic

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