To find the common ratio, r, we divide any term
by its preceding term.

So we can either divide the second term 12 by the 
first term 36 and get  which reduces to
.  So 

or

we can divide the third term 4 by the second term 12 
and get  which also reduces to .  So 
either way we get .

I'll do it two ways, the first way we'll do it will not 
be acceptable to your teacher, but it gets the right 
answer, so you'll know when you get it right.

The first way is to write the terms out till you get
to  by multiplying by  each time:

36, 12, 4, 

So to get the next term we multiply 4 by  and get .
So far we have:

36, 12, 4, , 

To get the next term we multiply 4/3 by  and get .
So far we have:

36, 12, 4, , 

To get the next term we multiply 4/9 by  and get .
So that's it, we now have

36, 12, 4, , , 

Then we count the terms and see that there are 6.

But your teacher doesn't want you to do that in case there
might have been 100 terms! But at least we know the answer
is 6.

Here's what your teacher wants to to do:

The formula for the nth term of a geometric sequence is









Cross multiply:



Simplify by dividing both sides by 4:



Write  and  and  as 



Add exponents on the right:



Use the principle: If  and  and 
                   then 




Edwin