Recognise geometric growth and decay, use recurrence relations and the explicit rule for geometric sequences, and model compound and reducing situations.

What this dot point is asking

You must recognise constant-ratio (geometric) change, write both the recurrence and explicit rules, and apply them to growth and decay contexts.

Geometric sequences

A sequence is geometric if each term is the previous term times a fixed common ratio $r$. Constant ratio means exponential (geometric) change, in contrast to a constant difference, which is linear (arithmetic).

Here $r > 1$ gives growth and $0 < r < 1$ gives decay. The recurrence is best for "next from current" and calculator tables; the explicit rule lets you jump straight to any term.

Growth and decay rates

A growth rate of $R\%$ per period gives $r = 1 + \dfrac{R}{100}$. A decay rate of $R\%$ per period gives $r = 1 - \dfrac{R}{100}$. For example, $5\%$ growth gives $r = 1.05$; $8\%$ decay gives $r = 0.92$.

Linear versus geometric

Decide which model fits before calculating. Linear change adds a constant amount each step (flat-rate depreciation, simple interest). Geometric change multiplies by a constant ratio each step (reducing-balance depreciation, compound interest). Plotting helps: linear data lies on a straight line, geometric data curves.