# Geometric Sequences

## Key Questions

• 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 $r$ as follows:
1) $\frac{20}{4} = 5$
2) $\frac{100}{20} = 5$

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:

• $1 , 2 , 4 , 8 , \setminus \cdots$ - the common ratio is 2.

• $- 1 , 3 , - 9 , 27$ - the common ratio is -3.

• $7 , 1 , \frac{1}{7} , \frac{1}{49}$ - the common ratio is 1/7.