Compound Interest:
A = P * (1 + r/n)^(n * t)
A = final amount
P = initial amount
r = rate of interest
n = number of times interest is compounded per period
t = number of periods
This is pretty straightforward. With compound interest, we start with an initial amount (P) and interest is applied. Then interest is applied to the new amount, and we repeat until we're tired. For instance, suppose you have 100 dollars and it increases at 10% per year, and it's compounded once per year. How much will you have after 3 years?
A = 100 * (1 + 0.10/1)^(1 * 3) = 100 * (1.1)^3 = 100 * (1.331) = 133.1
What if it was compounded twice per year?
A = 100 * (1 + 0.10/2)^(2 * 3) = 100 * (1 + 0.05)^6 = 100 * 1.05^6 = 100 * 1.340095640625 = 134.01
The more often it's compounded, the more interest is earned in a given period. There is an upper limit to this compounding, but that won't really help us for our problems, so there's no need to go further with that.
Geometric series.
Geometric series are series of numbers where each term is separated by a common ratio. For instance: 1 , 2 , 4 , 8 , 16 , 32 is a geometric series with a common ratio of 2 (each term is twice the amount of the term to its left and half the amount of the term to its right).
Now, geometric series are pretty special in that there's a trick for adding all of the terms together quickly and easily. It takes a little bit of algebra, but it can be done.
Let each term of a series be denoted by this formula: t[n] = a * r^n
Then the series would be:
a * r , a * r^2 , a * r^3 , a * r^4 , a * r^5
For instance 1 , 2 , 4 , 8 , 16 would be: (1/2) * 2^1 , (1/2) * 2^2 , (1/2) * 2^3 , (1/2) * 2^4 , (1/2) * 2^5
Let's add the first p terms of a series together. We'll call the sum S
S = ar + ar^2 + ar^3 + ar^4 + ... + ar^p
Now, multiply each side by the common ratio of r