If the initial term of a geometric progression is a and the common quotient of successive members is r, then the n-th term of the sequence is given by a·rn, n = 0, 1, 2, ...
The sum of the numbers in a geometric progression is called a geometric series. A convenient formula for geometric series is available.
One ordinarily distinguishes between two kinds of progressions, arithmetical and geometrical, corresponding to the proportions called arithmetical and geometrical. But the word 'proportion' seems rather inappropriate as applied to arithmetical proportion.
The idea of proportion is already well established by usage and it corresponds solely to what is called geometrical proportion; when we say generally that one thing is proportional to another, we understand by proportion equality of ratios only, as in geometrical proportion, and never equality of differences as in arithmetical proportion.
One cannot see why the proportion called arithmetical is any more arithmetical than that which is called geometrical, nor why the latter is more geometrical than the former. On the contrary, the primitive idea of geometrical proportion is based on arithmetic, for the notion of ratios springs essentially from the consideration of numbers