Consider the mean formula of the Negative Binomial distribution:

I don't understand why the range for X in the summation formula starts from r. I've looked up several textbooks and they seem to be treating this as obvious - but this is not obvious to me.

(The mean formula is just an example of one of the functions of NegBin: I have the same question about any other (variance, moment generating function) formulas for NegBin which include the same range of X.)

Here is the most relevant comment on the range of R I found (internet):

"Let X be the random variable which is the number of trials up to and including r-th success. This means that the range of X is the set {r, r+1, r+2,...}."

This sounds couter-intuitive to me, if X is the number of trials UP to r, then why do we start counting from r? why not start from x=1 (first trial) until x=r (the last successful outcome we need to 'complete' the r)?

Appreciate your help.