Financial and other sequence applications | Year 11

01 / LEARN

Understand the idea.

Choose how the change works

Choose a model from the type of change: a fixed amount gives arithmetic behaviour, a fixed percentage gives geometric behaviour, and a percentage combined with a fixed payment gives an affine recurrence. Identify the changing quantity before choosing a formula.

A value after several periods is a term. A total of successive payments is a sum. State the starting time: an initial value at $n = 0$ undergoes $n$ changes by time $n$ , while the first payment at index one is already included.

Assumptions that matter

Convert a percentage into its multiplier: an increase by rate $p$ uses $1 + p$ , while a decrease uses $1 - p$ . Match the rate to the period. An effective annual rate cannot simply be treated as the same monthly rate.

For payment models, mark whether deposits occur before or after interest. Interpret the answer under the given assumptions, including fixed rates and any omitted fees or taxes. Retain full precision until the requested final rounding.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Depreciation as a term

A machine costs AUD 24,000 and loses 15% of its value yearly. Find its value after three years.

Worked solution

1
Find the fraction retained each year.
$r = 1 - 0.15 = 0.85$
Losing fifteen percent leaves eighty-five percent of the previous value.
2
Model the value at the stated time.
$V_{0} = 24000, V_{3} = 24000 {(0.85)}^{3}$
Three years means three successive percentage reductions from the initial value.
3
Calculate the remaining value.
$V_{3} = AUD 14739$
The question asks for a single value after three years, rather than a total of yearly values.

Answer

$14, 739$ .

EXAMPLE 02

Increasing weekly deposits

A student saves AUD 12 in week 1 and increases the weekly amount by AUD 3. Find total deposits over 10 weeks.

Worked solution

1
Identify the deposit sequence.
$a = 12, d = 3, n = 10$
The amount deposited increases by a fixed three dollars per week.
2
Find the final weekly payment.
$u_{10} = 12 + 9 (3) = 39$
There are nine increases between weeks one and ten.
3
Add all ten deposits.
$S_{10} = \frac{10 (12 + 39)}{2} = 255$
An arithmetic total uses the average of the first and last payments.
4
State the total with its unit.
$Total deposits = AUD 255$
The result is the sum paid over ten weeks; no interest is specified.

Answer

$255$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

An initial value of AUD 500 rises by AUD 40 each month. Give its value after $n$ months.

Show answer & reasoning

$V_{n} = 500 + 40 n$ .

The initial value is at $n = 0$ and each month adds AUD 40. After $n$ months there have been $n$ additions, giving $V_{n} = 500 + 40 n$ for whole-number $n \geq 0$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

A AUD 1500 balance grows 3% yearly with no payments. Find its value after two years.
2 marks
A theatre begins with 18 seats in row 1 and adds two per row. Find total seats in 12 rows.
2 marks
A bouncing object travels 8 m on its first horizontal bounce and 75% as far on each later bounce. Find its ideal total.
2 marks
AUD 100 is deposited at the start of each year for three years at 10% annual interest. Find the balance at the end of year 3.
2 marks
A AUD 600 debt grows 5% before each AUD 150 repayment. Find the balance after two repayments.
2 marks
A tree-count model assumes 20% growth yearly from 100 trees forever. Give its rule and one limitation.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A AUD 1500 balance grows 3% yearly with no payments. Find its value after two years.
2 marks
$1500 {(1.03)}^{2} = 1591.35$ .
$1591.35$ .
A theatre begins with 18 seats in row 1 and adds two per row. Find total seats in 12 rows.
2 marks
$\frac{12 [36 + 11 (2)]}{2} = 348$ .
$348$ .
A bouncing object travels 8 m on its first horizontal bounce and 75% as far on each later bounce. Find its ideal total.
2 marks
$\frac{8}{1 - 0.75} = 32$ .
$32 m$ .
AUD 100 is deposited at the start of each year for three years at 10% annual interest. Find the balance at the end of year 3.
2 marks
$100 [{(1.1)}^{3} + {(1.1)}^{2} + 1.1] = 364.1$ .
$364.10$ .
A AUD 600 debt grows 5% before each AUD 150 repayment. Find the balance after two repayments.
2 marks
$B_{1} = 480$ and $B_{2} = 1.05 (480) - 150 = 354$ .
$354$ .
A tree-count model assumes 20% growth yearly from 100 trees forever. Give its rule and one limitation.
3 marks
A constant percentage gives the geometric model, but indefinite growth is physically implausible.
$T_{n} = 100 {(1.2)}^{n}$ ; finite land/resources eventually limit growth.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics sequence and series content with introductory induction, regrouped into an original school-sized chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Choose how the change works

Assumptions that matter

See the method in action.

Depreciation as a term

Find the fraction retained each year.

Model the value at the stated time.

Calculate the remaining value.

Increasing weekly deposits

Identify the deposit sequence.

Find the final weekly payment.

Add all ten deposits.

State the total with its unit.

Try it yourself.

Check your understanding

Keep practising