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

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A $2,000 investment earns 5% simple interest per year. Which value is used to calculate every year’s interest?

Sam invests $43,810 in an investment account held on a child's behalf, which pays simple interest at 3% p.a.

This is modelled with a recurrence relation below, where \(V_n\) is the balance after \(n\) years:

V_0 = \$43{,}810

V_{n+1} = V_n + \$1{,}314.30

3 marks

Complete the table below, showing the balance at the end of each year

Year	Balance
0	$43,810
1	?
2	?
3	?

Balance after 1 year ($) Balance after 2 years ($) Balance after 3 years ($)

Ethan takes out an education loan of $11,230, charged simple interest at 13.6% p.a. No repayments or fees apply

(a)1 mark

Find the amount of interest for one year

Interest per year ($)

(b)3 marks

Complete the table below, showing the amount owed at the end of each year

Year	Amount owed
0	$11,230
1	?
2	?
3	?

Amount owed after 1 year ($) Amount owed after 2 years ($) Amount owed after 3 years ($)

(c)1 mark

What is the total interest charged over the 3 years?

Total interest ($)

A high-interest savings account pays simple interest at 5.6% p.a. Mohammed opens one with $6,930

(a)1 mark

How much interest is earned each year?

Interest per year ($)

(b)4 marks

Complete the table below, showing the balance at the end of each year

Year	Balance
0	$6,930
1	?
2	?
3	?
4	?

Balance after 1 year ($) Balance after 2 years ($) Balance after 3 years ($) Balance after 4 years ($)

(c)1 mark

What is the total interest earned over the 4 years?

Total interest ($)

(d)1 mark

This situation is modelled by a recurrence relation. What is the starting value, \(V_0\)?

V0 ($)

(e)1 mark

Each new value equals the previous value with the per-year amount:

subtracted

added

(f)1 mark

What is that per-year amount, \(D\)?

D ($)

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, Calculator allowedBuild a worksheet

Simple interest

The same dollar amount is added for each equal period

Principal P is the original amount, equal to V₀
Yearly interest r is the yearly percentage number. D is the yearly dollar interest, calculated from the original principal \(\displaystyle D=Pr/100\)
Balance Add n equal amounts to the start \(\displaystyle V_n=P+nD\)
Total interest Exclude the principal \(\displaystyle I_n=nD\)

Worked example

An investment of $2,500 earns 4% simple interest per year. Find the interest and balance after three years

\(\displaystyle D=2500\times4/100=100\)Calculate the fixed yearly interest
\(\displaystyle I_3=3\times100=300\)Three years gives $300 interest
\(\displaystyle V_3=2500+300=2800\)Add the principal for the balance

Common mistakes

Using the changing balance

✕\(\displaystyle D=0.04V_n\)✓\(\displaystyle D=0.04P\)

Simple interest uses the original principal

Balance instead of interest

✕\(\displaystyle I_3=2800\)✓\(\displaystyle I_3=300\)

The balance includes the original $2,500

Difficulty levels

introIterate a supplied investment relation

standardCalculate interest and balances for loans and investments

challengeIdentify the initial value, operation and fixed amount

examChoose the correct balance

Questions about this topic

Does simple interest earn interest on interest?+

No. Only the original principal is used

What does p.a. mean?+

Per annum, meaning per year

Why does a loan grow?+

These models add interest without repayments or fees. A repayment loan needs those cash flows included

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 3: Sequences and finance

In this chapter

Identify the behaviour of a sequence Find the rule for a sequence and give the next term 3A Sequence patterns and rules: Generate the terms of a sequence from a starting value and a rule 3B Terms and recurrence relations: Generate the terms of a sequence from a starting value and a rule Name and number the terms of a sequence 3B Terms and recurrence relations: Write a recurrence relation for a sequence Find the common difference and the next term of an arithmetic sequence Decide whether a sequence is arithmetic, geometric or neither Tabulate a sequence and graph its terms 3D Arithmetic recurrence and terms: Write a recurrence relation for a sequence Find term n of an arithmetic sequence using the rule 3D Arithmetic recurrence and terms: Find when a sequence first reaches or passes a given value Model a simple-interest investment or loan with a recurrence relation Model flat-rate depreciation with a recurrence relation Model unit-cost depreciation with a recurrence relation Use the rule for term n of a linear growth or decay model Read the financial parameters out of a linear recurrence relation 3E Linear growth and decay in finance: Find when a sequence first reaches or passes a given value Build and compare linear financial models Find the common ratio and the next term of a geometric sequence Tabulate and interpret discrete geometric graphs Distinguish arithmetic and geometric patterns Write and interpret geometric recurrence rules Write an explicit geometric sequence rule Find term n of a geometric sequence using the rule Find the first geometric threshold index 3H Compound interest and depreciation: Find when a sequence first reaches or passes a given value Model a compound-interest investment or loan with a recurrence relation Write a recurrence relation for a loan or investment with a given compounding period Construct complete compound-interest recurrence models Find the first financial threshold period Construct and use reducing-balance depreciation models 3I Geometric models and finance: Find when a sequence first reaches or passes a given value Use the rule for term n of a compound-interest or reducing-balance model Find the interest on a credit-card debt compounding daily Compare the total cost of ways of paying, or the return on ways of investing Find the effect of inflation on a price Find the purchasing power of an amount after inflation