Find the number of terms in the geometric progression 7,21, 45927.

What lists are the easiest to count? If you were given a list like 1, 2, 3, 4, ... , 20, it would be pretty easy to determine that there were 20 terms in the sequence. We can transform most sequences to that form to make our job of counting terms a lot easier.

First, we notice that the terms all seem to be multiples of 7. We can start to simplify our list by dividing each term by 7:



Looking at our new list, you might notice that our numbers all seem to be powers of 3. , and you can verify that 

We can now transform our list again, replacing each term with its corresponding power:



Now, we simply add 1 to each term and:



Voila! There are 9 terms in the sequence.