Well if we just let X ~ Geo(p) then X can represent the outcome from your first paragraph. Let Y be a random variable associated with the process you're interested in. It's easy to see that Y = N*X. So we can just use facts like if Z is a random variable then E[c*Z] = c*E[Z] for any real value of c and SD[c*Z] = sqrt(c)*SD[Z]