When a population of individuals, dollars in a bank account or other collection of discrete items grows by a constant ratio over discrete periods, it is said to have a geometric growth rate. Understanding how geometric growth rates work and calculating them is very useful when looking at things like compound interest or ecology.
What Is a Geometric Growth Rate?
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According to several patterns, a population (or other collection or set) can grow (or shrink/decay). The simplest is called arithmetic or linear growth: when a constant number A is added to the existing population at each discrete time interval. For example, you might put $20 in a savings account every week. The first week you would have $20; the next week, you'd have $40, then $60 and so on. You can find the constant A by subtracting any two consecutive terms in the sequence: $60 - $40 = $40 - $20 = $20.
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But what if instead of growing by a fixed, constant number, the population grew by a percentage? This can also be seen as growing by a constant ratio. For example, imagine an initial population of 1,000 birds grows by 10 percent every year. You would start with 1,000 birds, then at the end of the first year there would be 1,000 + (1,000 * 0.10) = 1,000 + 100 = 1,100 birds. If this was linear growth, then there would be 1,100 + 100 = 1,200 birds at the end of the second year.
However, with geometric growth, you would actually be looking at 10 percent of 1,100 now, which would be 110, for a total of 1,210 birds. Those extra 10 birds come from the geometric nature of the growth. The following year, instead of 1,300 birds, you'd have 1,331 birds, now with 31 more birds than arithmetic growth would yield. The number of "extra" birds will keep getting larger, as each time, you're looking at 10 percent of a larger and larger term. In this case, 10 percent is the geometric growth rate of this bird population.
The Geometric Growth Rate Formula
How do you determine the geometric growth rate of a population? If you know the growth rate, how do you predict the population at a certain time in the future? If you're looking at linear growth, this is very straightforward. The linear constant is simply the difference between any two consecutive terms. If you only have the first term and the term after a certain time t, then you subtract the initial population from the final population and divide by the time (here, in years) to obtain the constant. In general, the population after t years is A * t * Initial Population.
Geometric growth is more complicated. Since you know that the ratio between terms is constant, you can find that ratio if you have any two consecutive terms. In the example, take the populations at the end of Year 2 (1,210) and Year 3 (1,331). Dividing the Year 3 population by the Year 2 population gives us 1,331 / 1,210 = 1.10. This is equal to 1 + 0.10, where 0.10 is the growth rate specified for the example. The 1 is added to account for including the preceding year's population in the calculation; in other words, the next year's population is equal to 110 percent (1.10x) of the previous year's population. The growth rate is commonly written as r, and the term (1 + r) is referred to as the growth multiplier, as explained by Lumen Learning.
More on Geometric Growth
But what if you don't have two consecutive terms, but only the starting population and the population after t years? First, you must observe that the population after t years (Pt) follows the formula Pt = P0 * (1 + r) ^ t, where P0 is the initial population. This formula is given, for example, in a United Nations Stats Brief on growth rates, since it's a useful measure when looking at economic or population growth.
This formula works because every term multiplies the previous term by the growth multiplier, so to find any term, you just keep multiplying by (1 + r), which is the same as raising (1 + r) to an equivalent power. With a little algebra, you can rearrange the geometric growth equation to find r:
Pt / P0 = (1 + r) ^ t
1 + r = (Pt / P0) ^ (1/t)
r = [(Pt / P0) ^ (1/t)] -1
If you recall that an exponent of 1/x is equivalent to taking the xth root, you can understand that an exponent of 1/2 is equivalent to taking the square root. This formula can be put into a scientific calculator, Excel or any other computational aid that can do custom exponents.
Using a Calculator for Geometric Growth
You can apply the above formula to the example as it would be entered in a calculator, using the initial population of 1,000 and the population at t = 3, 1,331. First, divide Pt by P0. Enter 1331 and hit the division key followed by 1000 and the equals sign to return 1.331.
Next, find 1/t. Divide 1 by 3 (the number of periods) to obtain 0.3333 repeating. If your calculator has a memory function, it's a good idea to store this number.
Then, find the calculator's exponentiation function, often labeled as x with a superscript of y. Enter 1.331, hit the exponentiation function, and enter 0.33333 or your stored value from above. Hit the equals sign to obtain something like 1.099999 (the exact decimal will depend on how precisely you entered the 1/3 number). Subtracting 1 and rounding gives you the correct 0.10 or 10 percent growth rate.
Geometric and Exponential Growth in Biology and Economics
In the growing bird population example, what would happen if you checked on the population at some intermediate point in the year? Or, if the savings account from the first example had an interest rate, how could you check the growth of your account? While these scenarios may seem unrelated, they can be modeled similarly.
Geometric growth is a special case of what is known as exponential growth. Looking at the savings account example, imagine it has a 10 percent interest rate that is added back into the account, or compounded, at the end of every year. You can see how this is exactly the same model as the growing bird population. If you start with $1,000, at the end of 3 years, you would have $1,331. But what if the interest was compounded monthly? That means you would be invoking the growth ratio much more often. The compound growth formula takes this into account: FV = V0 * (1 + r/n) ^ (t * n), where FV is the final value, V0 is the initial value, t is the length of time in years and n is the number of compounding periods per year.
The geometric growth formula used previously is actually the case where n =1, or there is one compounding period per year. Applying this formula to the savings account, compounding the interest monthly would give you $1,000 * (1 + 0.10/12) ^ (3 * 12) = $1,000 * (1 + 0.008333) ^ 36 = $1,348.18. Compounding monthly netted over $17 extra after 3 years.
Consider also: How to Calculate an Expected Growth Rate Using Constant Growth