Getting a math problem out of my head

I am up in the middle of the night figuring out factorial like addition in my head. When people talk about factorials they mean multiplication and notate it n! so what do I mean by factorial like addition?

In the middle of the night I decided to do some pushups in a reverse pyramid from 10. So I did 10 reps, then 9, then 8, then 7 and so on until I got to 1. How many did I do in all? 10+9+8+7+6+5+4+3+2+1 = 55. While I'm sure mathematicians have figured this out in the past, I still wanted to work out the pattern on my own. After a bit of trial and error here is what I came up with.

n^2/2 + n/2 where n is the number I start with.

Then I thought about a similar thing I do by 5s. I trained myself to be able to do a decent amount of not so decent crunches by pyramiding down by 5. So I'd start at say 50 reps, then 45 and so on until I got to 5. The previous way to work it out didn't match so I kept spinning my wheels to figure out something that works for both(and other numbers as well).

n^2/d*2 + n/2 where n is the number I start with and d is the amount I decrement by each time.
So my example of 50, 45, et cetera looks like this:
50^2/5*2 + 50/2
2500/10 + 25
250 + 25
275 = 50 + 45 + 40 + 35 + 30 + 25 + 20 + 15 + 10 + 5

From my minuscule amount of adding numbers in my head this works for decrementing by 1, 2, 3, 5, 10 quite well as long as you start with a multiple of the decrement amount, thus ending on the decrement amount. For other values it seems like you just need to round up your solution, though I haven't really tried with a lot of numbers to verify.

So, I might not have totally figured this out but I got somewhere so that I can answer this math problem quicker in the future. Most likely it will be in the same context of reverse-pyramiding reps. However, I'd like to know the actual name for what this describes in mathematic terms if anyone knows what that is.

Its getting real...funny

My farewell show at Comedysportz San Jose