Population growth formula
The geometric growth model is N = N₀ × (1 + r)^t, where N₀ is the starting population, r is the growth rate per period (as a decimal), and t is the number of periods. Each period the population is multiplied by (1 + r), giving compound exponential growth.
The exponent is where the power of the model lies: (1 + r) applied t times in succession is the same as raising it to the power t, so small differences in r produce very large differences over long spans. Exponential growth is repeated multiplication, which is what the exponent calculator does directly.
How to use this calculator
Enter the initial population, the growth rate as a percentage, and the number of periods (years, generations, etc.). The tool compounds the rate over time and rounds to a whole population. To find how long until the population doubles, use the doubling time calculator.
The rate and the period must match each other. A 5% annual rate paired with 10 years is fine; the same 5% paired with 10 months would need a monthly rate instead.
Worked example
Start with 1,000 individuals growing 5% per year for 10 years: N = 1000 × (1.05)^10 = 1000 × 1.6289 ≈ 1,629 individuals, an increase of about 629.
Treat the output as a projection, not a forecast. It assumes the growth rate stays fixed and that nothing limits expansion. Real populations run into food, space and disease limits, so the curve flattens into an S shape and this model overstates the true figure the further out you push it. Year-on-year change is often quoted as a percentage, and the percentage increase calculator converts between the two.
Frequently asked questions
- Is this exponential or logistic growth?
- This is exponential (geometric) growth, which assumes unlimited resources. Real populations eventually slow as they approach a carrying capacity (logistic growth).
- What does r mean here?
- r is the growth rate per period as a percentage. A 5% rate means the population multiplies by 1.05 each period.
- Can the rate be negative?
- Yes. A negative rate models a shrinking population; for example, −3% per year means the population is multiplied by 0.97 each period.
- What time unit should I use?
- Any consistent unit — years, days, or generations — as long as the growth rate is expressed per that same period.
- Why is my projection so much higher than adding 5% ten times?
- Because growth compounds. Adding 5% of the original 1,000 ten times would give 1,500, but each period's growth is calculated on the new, larger total, which is why the model returns about 1,629 instead.