Consider a piece of radioactive material. No matter what the size of the material, the proportion of it that will undergo radoactive decay in a given time period is the same. The half life of a radioactive element is the amount of time that it takes for half of a material made up of that element to = decay. If the half life of an element were one day then half of the material would remain radioactive after a day and a quarter of it after two days. After n days the amount of radioactive material remaining would be (1/2)n. This is an example of exponential decay. It is analogous to the traveler problem. Amount of material remaining corresponds to distance remaining and amount of material that has decayed corresponds to distance traveled. If we wrote an expression for the amount of material that decayed by adding the terms for each day we would again have a geometric series.
We can also have exponential growth, where the amount by which something increases is proportional to the total amount. The most familiar example is compound interest. In the traveler problem the distance taveled was 2/3 of the total remaining distance. We subtracted 2/3 from 1 to get 1/3 and the amount remaining after n time periods was (1/3)n. Now money is deposited in a bank at 5% interest. We add .05 to 1 and the amount of money after n years is increased by a factor of (1.05)n. If the initial deposit is $1 then the total amount of money at the start of the nth year would be (1.05)n-1 and so the interest earned in the nth year would then be .05*(1.05)n-1.
Let us do a derivation of
geometric series using this compound
example of exponential growth. We have:
(total amount of money at end of n years) - (total amount of interest earned (starting amount of $1).
(1.05)n - (.05 + .05*1.05 + .05*(1.05)2 + ... .05*(1.05)n-1) = 1.
Solving for the total amount of
interest and then dividing by .05 =
1 + 1.05 + (1.05)2 + ... .(1.05)n-1 = ((1.05)n - 1)/.05,
which is the formua for Sn(1.05).
Another example of exponential growth is the growth of an animal or = plant population that is not held in check by predation or limited resources. = The increase in population in any period of time is proportional to the size = of the population, leading to exponential growth.