Question

What is the common ratio of the geometric sequence 2, 6, 18, 54,...?

Answer 1

`3`

A geometric sequence has a common ratio, that is: the divider between any two nextdoor numbers:

You will see that `6//2=18//6=54//18=3`
Or in other words, we multiply by `3` to get to the next.
`2*3=6->6*3=18->18*3=54`

So we can predict that the next number will be `54*3=162`

If we call the first number `a` (in our case `2`) and the common ratio `r` (in our case `3`) then we can predict any number of the sequence. Term 10 will be `2` multiplied by `3` 9 (10-1) times.

In general
The `n`th term will be`=a.r^(n-1)`

Extra:
In most systems the 1st term is not counted in and called term-0.
The first 'real' term is the one after the first multiplication.

This changes the formula to `T_n=a_0.r^n`
(which is, in reality, the (n+1)th term).