Geometric Sequences
Key Questions

The common ratio of a geometric sequence, denoted by
#r# , is obtained by dividing a term by its preceding termconsidering the below geometric sequence:
#4 , 20 , 100# ...we can calculate
#r# as follows:
1)#20/4 = 5#
2)#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.
