Question

Find the number of terms in the geometric progression 7,21,63,..., 45927.

Answer 1

The rule is times 3

7 x 3 = 21
21 x 3 = 63

and so on until 45927

Answer 2

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:

$7,21,63,\dots,45927\rightarrow1,3,9,\dots,6561$

Looking at our new list, you might notice that our numbers all seem to be powers of 3. $1=3^0,3=3^1,9=3^2$, and you can verify that $6561=3^8$

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

$1,3,9,\dots,6561\rightarrow0,1,2,\dots,8$

Now, we simply add 1 to each term and:

$0,1,2,\dots,8\rightarrow1,2,3,\dots,9$

Voila! There are 9 terms in the sequence.