I think it would be better to define an arithmetic sequence as one with a constant difference and a geometric sequence as one with a constant ratio. This allows all the other properties of these sequences to be easily derived. This also allows more general types of sequences to be similarly defined (e.g., the differences grow linearly).

(I’m going through this and commenting when something strikes me.)

Your use of “series” to mean “partial sums of a sequence” seems odd to me.

Also, you do not need to consider even and odd n cases for the sum 1+2+…+n.
Let S = 1+2+…+(n-1)+n. Writing this in reverse order,
S = n+(n-1)+…+2+1. Adding the corresponding terms,
2S = (n+1)+(n+1)+…+(n+1)+(n+1) = n(n+1)
so S = n(n+1)/2.

Another way to get the sum of a geometric sequence is this:
If S = 1 + r + r^2 + … + r^(n-1) + r^n,
rS = r + r^2 + … + r^n + r^(n+1)
so S – rS = 1 – r^(n+1)
s0, if r \ne 1, S = (1 – r^(n+1))/(1-r).
If r = 1, S = n+1, of course.

This has the advantage of not having to know the result in advance
and can also be generalized to get the sum of more complex series
(such as 1 + 2r + 3r^2 + … + (n+1)r^n).

I think it would be better to define an arithmetic sequence as one with a constant difference and a geometric sequence as one with a constant ratio. This allows all the other properties of these sequences to be easily derived. This also allows more general types of sequences to be similarly defined (e.g., the differences grow linearly).

(I’m going through this and commenting when something strikes me.)

Your use of “series” to mean “partial sums of a sequence” seems odd to me.

Also, you do not need to consider even and odd n cases for the sum 1+2+…+n.

Let S = 1+2+…+(n-1)+n. Writing this in reverse order,

S = n+(n-1)+…+2+1. Adding the corresponding terms,

2S = (n+1)+(n+1)+…+(n+1)+(n+1) = n(n+1)

so S = n(n+1)/2.

Another way to get the sum of a geometric sequence is this:

If S = 1 + r + r^2 + … + r^(n-1) + r^n,

rS = r + r^2 + … + r^n + r^(n+1)

so S – rS = 1 – r^(n+1)

s0, if r \ne 1, S = (1 – r^(n+1))/(1-r).

If r = 1, S = n+1, of course.

This has the advantage of not having to know the result in advance

and can also be generalized to get the sum of more complex series

(such as 1 + 2r + 3r^2 + … + (n+1)r^n).