Question

Is the sequence geometric? if so, identify the common ratio 1/5,2/15,4/45,8/135,16/405,...

Answer 1

The answer is :

yes;
`(2)/(3)`

Answer 2

So have the sequence:
`(1)/(5) , (2)/(15) , (4)/(45) , (8)/(135) , (16)/(145) ,...`
To check if the sequence is geometric, we are going to find its common ratio; to do it we are going to use the formula:
`r= (a_(n) )/(a_(n-1))`
where

`r` is the common ratio

`a_(n)` is the current term in the sequence

`a_(n-1)` is the previous term in the sequence
In other words we are going to divide the current term by the previous term a few times, and we will to check if that ratio is the same:

For
`a_(n)= (2)/(15)` and
`a_(n-1)= (1)/(5)`:

`r= ( (2)/(15) )/( (1)/(5) )`

`r= (2)/(3)`

For
`a_(n)= (4)/(45)` and
`a_(n-1)= (2)/(15)`:

`r= ( (4)/(45) )/( (2)/(15) )`

`r= (2)/(3)`

For
`a_(n)= (8)/(135)` and
`a_(n-1)= (4)/(45)`:

`r= ( (8)/(135) )/( (4)/(45) )`

`r= (2)/(3)`
As you can see, we have a common ratio!

We can conclude that our sequence is a geometric sequence and its common ratio is
`(2)/(3)`