What is a Geometric Progression?

A geometric progression is a sequence of terms such that each term is multiplied by a constant value 'r' which is called the common ratio.

For example:

10, 20, 40, 80, 160 ... is a geometric progression where the common ratio, r = 2. (each term is multiplied by 2)

and

48, 24, 12, 6, 3, ... is a geometric progression where the common ration, r = 1/2 (each term is multiplied by 1/2)

In General

We can represent any geometric progression as

where 'a' is the first term and 'r' is the common ratio.

What is a Geometric Series?

If we add together the terms in a geometric progression then we have what is called a geometric series.

so

10 + 20 + 40 + 80 + 160 + ... is a geometric series with first term 'a' where a=10 and common ratio 'r', where r=2

In general

You are expected to know how to prove this summation formula so look at this video if you are not sure.

Proof of the Sum of the first n terms of a Geometric Series

Special Case - The sum to Infinity

If the common ratio 'r' lies between -1 and 1 (often written as -1<r<1 or |r|<1) then the geometric series will converge to a finite value.

It can be shown that

See the video for an explanation of the Sum to Infinity