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  interest 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 = gives:
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.