Mathematically, growth can occur in several different ways. For example, if you add one apple per second to a basket, the basket's weight increases arithmetically; the difference from one second to the next is the same. By contrast, a population of fish in a large lake increases geometrically; the growth from one week to the next is a fairly steady percentage, not a simple difference. To determine what the percentage increases are when growth is geometric, you can use the exponential functions of a scientific calculator.

## Step 1

Divide the final value by the initial value on the calculator by entering the final value, pressing the division sign, entering the original value and pushing the equals sign. For example, if the original population equals 150,000 and the final population equals 153,000, you would enter 153,000 and push the division sign, then enter 150,000 and push the equals sign. The result is 1.02.

## Step 2

Divide 1 by the number of years the growth takes place. Enter 1 and push the division sign, then enter the number of years and push the equals sign. For example, if the growth takes place over two years, you would enter 1 and push the division sign, then enter 2 and push the equals sign. The result is 0.5.

## Step 3

Raise the result from Step 1 to the power of the result from Step 2. Enter the Step 1 result and push the exponent sign, then enter the Step 2 result and push the equals sign. In this example, you would enter 1.02 and push the exponent sign, then enter 0.5 and push the equals sign. The result is 1.009950494.

## Step 4

Subtract 1 from the Step 3 result. Enter the Step 3 result and push the minus key, then enter 1 and push the equals key. In this example, you would enter 1.009950494 and push the subtraction sign, then enter 1 and push the equals sign. The result is 0.009950494.

## Step 5

Multiply the Step 4 result by 100 to find the annual geometric growth rate expressed as a percentage. Enter the Step 4 result and push the multiplication sign, then enter 100 and push the equals sign. Completing the example, you would enter 0.009950494 and push the multiplication sign, then enter 100 and push the equals sign. The result is the annual growth rate: about 0.995 percent.