Geometric Progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. A geometric sequence can be finite or infinite.

The n-th term an of a geometric sequence is given by:

$$a_{n}=a_{0}r^{n-1}$$

where,

a0— initial value;
r— common ratio;
n— 1, 2, 3 ...

Sum of n terms:

$$S_{n}=\sum_{k=0}^{n-1}a_{0}r^{k}=\frac{a_{0}(r^{n}-1)}{r-1}=\frac{a_{0}(1-r^{n})}{1-r}$$

Decreasing geometric progression

If the absolute value of the common ratio r is less than one, the geometric progression is called a decreasing geometric progression.

$$S=\frac{a_{0}}{1-r}$$