So we know the general formula for a sum of an arithmetic series is
Where is the number of terms in the series and is the starting number
The problem with this formula though is that it only handles when
So what if we want the formula when
Let's say that for now and so we have an answer to check our equation with.
This sum is almost the same as
The second sum can be worked out using the formula
And the first one is just
Since can be represented as , we can represent our equation as
And since both of those can be represented using the formula we can rewrite it as
Rewriting this using and gives us
Lets test it on our example
So it works (at least for and ), feel free to test it for yourself.