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:
[tex]7,21,63,\dots,45927\rightarrow1,3,9,\dots,6561[/tex]
Looking at our new list, you might notice that our numbers all seem to be powers of 3. [tex]1=3^0,3=3^1,9=3^2[/tex], and you can verify that [tex]6561=3^8[/tex]
We can now transform our list again, replacing each term with its corresponding power:
[tex]1,3,9,\dots,6561\rightarrow0,1,2,\dots,8[/tex]
Now, we simply add 1 to each term and:
[tex]0,1,2,\dots,8\rightarrow1,2,3,\dots,9[/tex]
Voila! There are 9 terms in the sequence.
