E1.17 Exponential growth and decay

Learning objectives

E1.17.1Use exponential growth and decay• Use exponential growth and decay. e.g. depreciation, population change. Knowledge of e is not required.

Exponential change applies the same percentage multiplier to the current amount in every equal time period. Because the amount changes each time, the numerical increase or decrease is not constant.

$A_n=A_0m^n$

Change per period	Multiplier $m$	Model after $n$ periods
growth by $r\%$	$1+\dfrac{r}{100}$	$A_n=A_0\left(1+\dfrac{r}{100}\right)^n$
decay by $r\%$	$1-\dfrac{r}{100}$	$A_n=A_0\left(1-\dfrac{r}{100}\right)^n$

A population of 250 000 decreases by 1.7% each year. After 5 years, $A_5=250000(0.983)^5=229460.9\ldots$ , so the population is 229 500 to the nearest hundred. The exponent is 5 because the multiplier is applied five times.

To recover an earlier amount, divide by the complete multiplier power. If a car is worth 6269.40 dollars after 3 years of 10% annual decay, its earlier value was $6269.40\div0.9^3=8600$ dollars.

For a missing rate, first isolate the multiplier: if 550 grows to 736 in 5 years, $m=\sqrt[5]{736/550}=1.06$ , so the growth rate is 6%. For the first whole period above or below a target, evaluate consecutive integer powers and choose the first one that crosses the target.

Use the number of percentage-change periods, not merely the difference between printed year numbers without checking the endpoints. Keep full calculator precision until the requested rounding, and do not replace repeated percentage change with simple change $A_0(1+nr/100)$ . Knowledge of $e$ is not required here.

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