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 S_{n}(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.