Definition:

By a geometric progression of m terms, we mean a finite sequence of the form

The real number a is called the first term of the geometric progression, and the real number r is called the ratio of the geometric progression.

Example 1:

Consider the finite sequence of numbers

4, 8, 16, 32, 64, 128, 256, 512, 1024

This sequence has the property that the ratio between successive terms is constant and equal to 2.

Here we have: a = 4; r = 2.

k-th term of the geometric progression:

the k-th term of the geometric progression is equal to

Sum of an geometric progression:

The sum of the m terms of a geometric progression  is equal to

Example 2:

Consider the geometric sequence 

Here we have: a = 1, r =

The sum of the first m terms is equal to

This is very close to 2 when m is very large.