What is wrong ratio for decay?

A 15\% loss means multiplying by 0.85, not 0.15. The ratio is what remains, not what is lost.

WAMathematics ApplicationsSyllabus dot point

How do we model situations that change by a constant percentage each step?

Recognise geometric sequences, use the recursive and explicit rules with the common ratio, and apply them to growth and decay.

How to identify a geometric sequence by its common ratio, use the recursive and explicit term rules, and apply the model to percentage growth and decay such as populations and reducing-balance depreciation.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The common ratio
Where geometric sequences appear
Geometric versus arithmetic

What this dot point is asking

You must recognise geometric behaviour, find the common ratio, use both the recursive and explicit rules, and apply them to percentage growth and decay.

The common ratio

A sequence is geometric when each term is a fixed multiple of the one before, the common ratio $r$ . You find $r$ by dividing any term by the previous one.

If $r > 1$ the sequence grows; if $0 < r < 1$ it decays towards zero. A percentage change converts to a ratio: a $5\%$ rise gives $r = 1.05$ , a $5\%$ fall gives $r = 0.95$ .

Where geometric sequences appear

The geometric model fits any quantity changing by a constant percentage each period.

Compound interest. The balance is multiplied by $(1 + i)$ each period, so it grows geometrically.
Reducing-balance depreciation. The asset keeps a fixed percentage of its value each year, multiplying by a ratio below $1$ .
Populations. A population growing at a constant percentage rate follows a geometric sequence.

Worked example

Reducing-balance depreciation as a geometric sequence

A \ $24\,000 machine depreciates at$ 15% $of its value each year. Keeping$ 85% $each year gives the common ratio$ r = 0.85 $, with$ a = 24,000$.

Recursive rule: $t_{n+1} = 0.85\,t_n$ , $t_0 = 24\,000$ .

Explicit rule for the value after $n$ years: $t_n = 24\,000 \times 0.85^{\,n}$ .

Value after $4$ years: $t_4 = 24\,000 \times 0.85^4 = 24\,000 \times 0.52200625 = 12\,528.15$ , that is about $12,528.

Each year the value is multiplied by $0.85$ , so the dollar drop shrinks over time, the curved decline typical of reducing-balance depreciation.

Geometric versus arithmetic

An arithmetic sequence adds a constant and plots as a straight line; a geometric sequence multiplies by a constant and plots as a curve. Recognising which model applies, constant amount or constant percentage, is the first decision in any growth or decay problem.