Population Growth — Definition, Formula & Examples

Population growth is an application of exponential functions where a population increases by a fixed percentage over equal time intervals. The starting population is multiplied repeatedly by a growth factor, causing the total to rise faster and faster.

A population undergoing exponential growth is modeled by $P(t) = P_0(1 + r)^t$ , where $P_0$ is the initial population, $r$ is the constant growth rate per time period, and $t$ is the number of time periods elapsed. This assumes unlimited resources and a fixed proportional increase each period.

Key Formula

P(t) = P_0(1 + r)^t

Where:

$P(t)$ = Population after t time periods
$P_0$ = Initial population at time t = 0
$r$ = Growth rate per time period (as a decimal)
$t$ = Number of time periods

How It Works

You start with a known population

P_0

and a growth rate

r

expressed as a decimal. For each time period, the population is multiplied by the growth factor

(1 + r)

. To find the population after

t

periods, raise the growth factor to the power

t

and multiply by

P_0

. Because the base

(1 + r)

is greater than 1, the function increases—slowly at first, then dramatically. If you need to find how long it takes the population to reach a target, set

P(t)

equal to that target and solve using logarithms.

Worked Example

Problem: A town has a population of 20,000 and grows at 3% per year. What is the population after 10 years?

Identify values: The initial population is 20,000 and the annual growth rate is 3%, so r = 0.03 and t = 10.

P_0 = 20{,}000,\quad r = 0.03,\quad t = 10

Substitute into the formula: Plug the values into the exponential growth model.

P(10) = 20{,}000(1 + 0.03)^{10} = 20{,}000(1.03)^{10}

Calculate: Evaluate the power and multiply.

(1.03)^{10} \approx 1.3439 \quad\Rightarrow\quad P(10) \approx 20{,}000 \times 1.3439 = 26{,}878

Answer: The population after 10 years is approximately 26,878.

Why It Matters

Population growth models appear in biology, ecology, economics, and public health whenever you need to project future numbers from a current trend. On the AP Precalculus and AP Calculus exams, these problems test your ability to set up and solve exponential equations, often requiring logarithms to find unknown time values.

Common Mistakes

Mistake: Using the growth rate r as the base instead of the growth factor (1 + r).

Correction: The base must be (1 + r). A 3% growth rate means you multiply by 1.03 each period, not by 0.03.