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.