Exponential growth — Mathematics, 14–17 years
How repeated percentage change creates curves that rise faster and faster. Mathematics, 14–17 years.
The idea
In linear growth, the same amount is added each time. In exponential growth, the same factor is used each time, so the increase becomes larger as the quantity grows. A population that multiplies by 1.2 each year follows the rule y = a × 1.2^n.
Why it matters
Adding a fixed number cannot describe situations where the current size changes the next change. Interest, infections and populations often grow by a percentage of what is already there. Exponential models were developed to make those repeated proportional changes calculable and comparable.
A worked example
A savings account starts with €500 and grows by 4% each year. The factor is 1.04, so after three years the amount is 500 × 1.04^3. Since 1.04^3 = 1.124864, the result is €562.43, rounded to the nearest cent.
The common trap
A reasonable mistake is to calculate 4% of the original €500 every year, giving 560 €. That is linear growth, because it adds €20 repeatedly. Exponential growth calculates each new percentage from the latest amount, so the later increases are slightly larger.
Where it appears
Banks use exponential models for compound interest, and scientists use them for early population or infection growth. A phone battery does not usually behave exponentially for its whole charge, so a model should not be stretched beyond the situation it describes. Always check the time range and the assumptions.
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