Are there any rules to judge a Sigma represents an arithmetic series or a geometric series or none of them?

[mathdaughter]

Are there any straightforward rules can be used to judge if a Sigma notation represents an arithmetic series or a geometric series without evaluation some of these items?

 

[Steven G]

An arithmetic sequence has a common difference between consecutive terms while a geometric series have a common ratio.

What form do you think the nth term of an arithmetic sequence would like (this term is what you will see in the sigma notation)?
Same question for a geometric sequence.

Do write out a few terms for each and then think about how you can see whether or not it is the sum of a AS of a GS from the sigma notation form.

 

[HallsofIvy]

The "rules" to use are the definitions of "arithmetic" and "geometric" series!

 

[Dr.Peterson]

If it is not obvious from the form of the term, you can subtract the nth term from the (n+1)st term (if it is a constant, you have an arithmetic series); or divide the (n+1)st term by the nth term (if this is constant, you have a geometric series).

This, as has been mentioned, is the definition, and definitions are one way to decide such questions. (In other cases, there are theorems that can help.)