A geometric sequence models population growth or disease spread when the quantity changes by the same percentage, or is multiplied by the same factor, during each equal time interval. The model is , where is the initial quantity and is the constant growth factor.
The mechanism is multiplicative. If is the population after time intervals, the recurrence relation is
Repeated substitution produces the explicit formula
This model assumes discrete, equally spaced intervals and a constant proportional rate. Each term depends on the previous term, so the sequence is geometric rather than linear. Graphically, gives exponential growth, gives exponential decay, and gives no change.
The value of the common ratio depends on the percentage change.
| Situation | Common ratio |
|---|---|
| Increase by per interval | |
| Decrease by per interval | |
| Quantity doubles per interval |
For example, suppose there are initially disease cases and the number of new cases increases by each week. Then and , so
After four weeks,
The model therefore predicts approximately new cases in week four. A population growing by annually would similarly use .
The same method can work backwards. If a population reaches after two years with , then . This distinguishes finding an initial value from forecasting a future value and shows how algebra rearranges the model.
A common misconception is to add the same numerical increase each period. That creates an arithmetic model. In geometric growth, the percentage is applied to the current quantity, so the numerical increase becomes larger each period.
Exam technique: This is IB Mathematics AA, SL 1.3: Geometric sequences and series. In a response, define the variables, identify and , write the model before calculating, and interpret the answer in context. For evaluation, note that real populations and epidemics may stop following a constant-ratio model because resources, immunity, or interventions change the growth rate.
