Arithmetical or geometrical sequence?

I have some problem in sequence, especially when I want to find the general formula of 'not arithmetic and also not geometric' sequences.

If I already knew that the sequence is arithmetic sequence, I can easily find the formula for arithmetic sequence, such that finding the general formula from 2,4,6,8..

We can use the formula Un=a+(n-1)d and we got Un= 2+(n-1)2= 2n

Another way to find the formula is using geometric sequence, for example we want to find the general formula of 3,6,12,24,...

We can use the formula Un= ar^(n-1), and we got: Un= 3.2^(n-1)

But, how about this sequence?

1/3, 3*sqrt 3, 27,...

Is there any way to find the general formula? I think I cannot use the formula from arithmetic and geometric sequences.

Please show me the way, so I can understand every steps.
Thanks

Answer

Hints: a sequence $\;a_1,a_2,...a_n,...\;$ is

$$\begin{align*}(1)&\;\;\text{Arithmetic, if}\;\;\forall\,n\in\Bbb N\;,\;\;d:=a_{n+1}-a_n\\{}\\

(2)&\;\;\;\text{Geometric, if}\;\;\forall\,n\in\Bbb N\;,\;\;q:=\frac{a_{n+1}}{a_n}\;\;\end{align*}\;\;\;\text{is a constant not depending on}\;\;n$$

In case (1), we call $\;d\;$ the constant difference, and in (2) we call $\;q\;$ the constant ratio.

Take now your sequence, look at it closely and when you suspect it is either Arithm. or Geom. (or its consecutive differences/ratios) apply the above and check .