Arithmetic and Geometric Progressions

Arithmetic Progressions

These are series of numbers e.g. 2, 6, 10, 14, 18 ... or 100, 97.5, 95, 92.5, 90, . . . . where a constant number is added to or subtracted from each term to form the next.
Assuming that the first term is a and that the number added/subtracted (the "common difference") is d, then the series is: a, a + d, a + 2d, a + 3d .... The nth term of the series is a + (n - 1)d.
The sum of the first n terms of the series is: 0.5n(2a + [n - 1]d).

Geometric Progressions

These are series of numbers, e.g. 2, 6, 18, 54, 162, ... or 100, -50, 25, -12.5, 6.25 ... where each term is multiplied/divided by a constant number (the "common ratio") to form the next. The number can be positive (in which case all the terms are the same sign) or negative (in which case alternate terms have alternate signs). The first term is usually called a. The common ratio is usually called r.
If r is outside the range -1 to 1, the terms of the series get bigger and bigger (even if they change sign), and the series diverges. If r is within the range -1 to 1, the terms get smaller and smaller (closer to 0) and the series converges.

The sum of the first n terms of a geometric progression is  a(1 - r n)/1 - r 

If r lies within the range -1 < r < 1, then the series has a sum to infinity (i.e. if you added up an infinite number of terms of the series, you would still get a finite number). An example is the series 1, 1/2, 1/4, 1/8, 1/16 ... etc. where a = 1 and r = 1/2. In this case, if you add the terms of the series, you get closer and closer to 2 without ever reaching it. If you could add up an infinite number of terms, then the sum would be 2.
To find the sum to infinity of the series, simply put n = ¥ in the equation above. Providing -1 < r < 1, the term r¥ will be 0, and the whole formula reduces to this:
a/1 - r