Model and analyse linear and geometric growth and decay using recurrence relations and rules.

What this dot point is asking

Growth and decay extend the financial recurrence ideas to any repeating process: populations, drug concentrations, radioactive material, or savings. The key question is whether each step adds a fixed amount (linear) or multiplies by a fixed factor (geometric).

Recurrence relations

A recurrence relation defines each term from the previous one and a starting value:

Linear: $t_{n+1} = t_n + d$, with $t_0$ given. Here $d > 0$ is growth, $d < 0$ is decay.

Geometric: $t_{n+1} = R\, t_n$, with $t_0$ given. Here $R > 1$ is growth, $0 < R < 1$ is decay.

The closed form lets you jump straight to any term without listing all the ones before it.

Recognising the model

If the differences between consecutive terms are constant, the data is linear. If the ratios between consecutive terms are constant, the data is geometric. Always test which one fits before choosing a rule.

Comparing the two models

Over the long run, geometric growth always overtakes linear growth, because multiplying repeatedly outpaces adding repeatedly. Geometric decay approaches zero but never quite reaches it, whereas linear decay hits zero (and would go negative if the rule kept applying).

A complete answer states the type of model, gives both the recurrence and the closed form, and rounds sensibly to the context (whole people for populations, two decimals for milligrams).