# How do you determine whether the sequence 3, 12, 21, 30,... is geometric and if it is, what is the common ratio?

Jul 17, 2017

It is not a geometric sequence and hence question of common ratio does not arise. It is actually arithmetic sequence and common difference is $9$.

#### Explanation:

A geometric sequence is one in which the ratio of each term to its preceding term is always constant. This ratio is called common ratio.

In the series, $\left\{3 , 12 , 21 , 30 , \ldots \ldots \ldots \ldots .\right\}$

we have $\frac{12}{3} = 4$, $\frac{21}{12} = \frac{7}{4}$, $\frac{30}{21} = \frac{10}{9}$

Hence it is not a geometric sequence.

However, we have $12 - 3 = 21 - 12 = 30 - 21 = 9$ and hence we have a common difference and hence it is an arithmetic sequence.