Exponential Growth Formula

Q: What do the symbols mean in the Exponential Growth formula?

$P_0$ initial value, $r$ growth rate, $t$ time.

The Formula

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

When to use: Exponential growth means the amount added each period is proportional to the current amount — the bigger it gets, the faster it grows, creating an accelerating curve.

Quick Example

Starting at 1, doubling each day: 1, 2, 4, 8, 16, 32, \ldots, 2^{30} \approx 10^9 after just 30 days. The growth looks slow at first, then explodes.

Notation

P_0 initial value, r growth rate, t time.

What This Formula Means

Exponential growth occurs when a quantity increases by a constant multiplicative factor over equal intervals.

Exponential growth means the amount added each period is proportional to the current amount — the bigger it gets, the faster it grows, creating an accelerating curve.

Formal View

A process is exponential when P(t+1)=kP(t) with constant k>1.

Worked Examples

Example 1

easy

A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 12 hours?

Solution

1
The exponential growth model is P(t) = P_0 \cdot 2^{t/d}, where d is the doubling time.
2
Substitute: P(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4.
3
P(12) = 500 \cdot 16 = 8{,}000.

Answer

8{,}000 \text{ bacteria}

Exponential growth occurs when a quantity increases by a constant percentage over equal time intervals. The doubling-time formula P(t) = P_0 \cdot 2^{t/d} is a special case of P(t) = P_0 \cdot b^t. In 12 hours, the population doubles 4 times: 500 \to 1000 \to 2000 \to 4000 \to 8000.

Example 2

medium

A city's population grows at 3\% per year. If the current population is 200{,}000, when will it reach 300{,}000?

Common Mistakes

Using additive instead of multiplicative change
Treating percent as a whole number (5 instead of 0.05)

Why This Formula Matters

Appears in finance, population models, epidemics, and technology trends.

Frequently Asked Questions

What is the Exponential Growth formula?

Exponential growth occurs when a quantity increases by a constant multiplicative factor over equal intervals.

How do you use the Exponential Growth formula?

Exponential growth means the amount added each period is proportional to the current amount — the bigger it gets, the faster it grows, creating an accelerating curve.