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

What this dot point is asking

You must recognise geometric behaviour, find the common ratio, use both forms of the rule, and apply the model 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 $R$. Test by dividing consecutive terms: if $\dfrac{t_2}{t_1} = \dfrac{t_3}{t_2} = \dots$, the sequence is geometric.

A growth of $r\%$ per period gives $R = 1 + \dfrac{r}{100}$; a decay of $r\%$ gives $R = 1 - \dfrac{r}{100}$. So $8\%$ growth is $R = 1.08$ and $12\%$ decay is $R = 0.88$.

Growth versus decay

The value of $R$ decides the whole shape of the model.

• $R > 1$: geometric growth, an upward curve that steepens (compound interest, populations).
• $0 < R < 1$: geometric decay, a downward curve that flattens towards zero (reducing-balance depreciation, radioactive decay).
• $R = 1$: a constant sequence; nothing changes.

Switching between the forms

Use the recursive rule to build a table or to let technology iterate; use the explicit rule $t_n = a R^{n-1}$ for a single far-off term. When given two terms, divide them to find $R$: if $t_2 = aR$ and $t_5 = aR^4$, then $\dfrac{t_5}{t_2} = R^3$, so $R = \sqrt[3]{\dfrac{t_5}{t_2}}$, and back-substitute for $a$.

Note the index convention: $t_n = a R^{n-1}$ uses $n-1$ when $t_1 = a$ is the first term. In finance the version $A_n = A_0 R^n$ is common, where $A_0$ is the starting value before any period. Decide which counting you are using and stay consistent.

Doubling and halving times

A common question asks how long a geometric model takes to double (for growth) or halve (for decay). Set the explicit rule equal to twice (or half) the start and solve by trial on the calculator. For $8\%$ annual growth, solve $1.08^n = 2$: testing gives $1.08^9 = 1.999$, so doubling takes about $9$ years. For a decay with $R = 0.85$, solve $0.85^n = 0.5$: $0.85^4 = 0.522$ and $0.85^5 = 0.444$, so the quantity halves during year $5$. Reporting the first whole period that crosses the target is the expected form of the answer.

Comparing arithmetic and geometric growth

Over a few periods, a steep arithmetic increase can stay ahead of a modest geometric one, but geometric growth always overtakes eventually because it compounds. This is the reason compound interest beats simple interest over time and why a small percentage population growth produces large absolute increases in later years. When a question gives both an "increases by a fixed amount" option and an "increases by a percentage" option, expect to model one as arithmetic and the other as geometric and compare them at a chosen period.