The common ratio of a geometric sequence, denoted by
#r#, is obtained by dividing a term by its preceding term
considering the below geometric sequence:
#4 , 20 , 100#...
we can calculate
#20/4 = 5#
#100/20 = 5#
so for the above mentioned geometric sequence the common ratio
# r = 5#
That lies in its very definition: there is a common ratio between successive terms of the geometric sequence. That is to say, a number is being multiplied to get each successive term, and this "number" is nothing but the common ratio. This means that you just have to check if there is a number being multiplied between successive terms.
If you also think about it, the common ratio is what you get when you divide a term by its predecessor. But you can probably already tell why you get the same number - it is because you have been multiplying the same number.
The "common ratio" can be anything, but it has been widely argued without a proper consensus that a sequence with a common ratio 0 or 1 should not be called geometric. But thinking about that argument is futile, as you would never, ever encounter a sequence such as that in questions. Even if you do, you can think for yourself.
Here are three examples:
#1, 2, 4, 8, \cdots#- the common ratio is 2.
#-1, 3, -9, 27#- the common ratio is -3.
#7,1, 1/7, 1/49#- the common ratio is 1/7.