We divide the 2nd term by the 1st term to find the common ratio 



To check we divide the 3rd term by the 2nd term to see if we get the 
same common ratio :

 

To double check we divide the 4th term by the 3rd term to see if we get the 
same common ratio :



Now that we are triple-sure that the common ratio ,

we will use the formula for the nth term:



to find out how many terms it has:

 is the last, or nth, term:
 is the first term
 is the common ratio.
Substituting:



Multiply both sides by 


 


Write  as 



Multiply both sides by 

Observe that , so substitute that:



Divide both sides by 



Subtract exponents on the left:





Since the right side is a power of
, it is either 1 or -1

But no power of 3 can be negative, so



And since the only power of 3 that gives 1
is the 0 power, i.e., 30=1, then
the exponent n-12 must equal 0.

n-12=0
   n=12.

So there are 12 terms.  So 



becomes:



write  as 



write  as 



Write  as :







Add exponents of  on top:





Subtract exponents of 3:





Edwin