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 — exponential growth multiplies by a factor each period, it does not add a fixed amount
Treating percent as a whole number — a 5% growth rate means r = 0.05, so the base is 1.05, not 1.5 or 5
Assuming early slow growth will stay slow — exponential functions look nearly linear at first but eventually explode; do not underestimate long-term exponential behavior

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.