Financial and other sequence applications | Year 11

01 / LEARN

The idea, step by step

Choose how the change works

A fixed increase per period suggests arithmetic terms; a fixed percentage change suggests geometric terms. A percentage change together with a fixed deposit or removal normally needs an affine recurrence.

Define the initial state and observation time. A balance at a particular time is not the same quantity as the total amount paid. Check whether the question needs a term, a finite sum or a limiting sum.

Assumptions that matter

Use an interest rate matching the time step. An annual effective rate cannot simply be reused as a monthly rate. An end-of-period payment has accumulated no interest at the instant it is paid.

Fees, taxes, inflation, changing rates and rounding may matter in reality. Explain the result under the supplied assumptions rather than presenting the model as a guaranteed financial outcome.

A common trap

Any question about interest requires an infinite sum.

Instead: Identify whether a finite balance, payment total or recurrence is needed; infinite sums are a distinct limiting model.

02 / WORKED EXAMPLES

See the thinking

Example 1

Depreciation as a term

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

The retention factor is 0.85.
$V_{3} = 24000 {(0.85)}^{3} = 14739$ . No sum is needed.

14, 739

Example 2

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.

$a = 12, d = 3, n = 10$ .
$S_{10} = 10 [24 + 9 (3)] / 2 = 255$ .

255

Check your understanding

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

Show answer

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

The monthly absolute increase is fixed.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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
$12 [36 + 11 (2)] / 2 = 348$ .
$348$ .
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
$8 / (1 - 0.75) = 32$ .
$32 m$ .
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 ↗

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