I like to think of the population size as the number of taxpayers who pay enough taxes to make a difference in a galactic economy. Right now, in 2001, I think Earth would be hard-pressed to find even a hundred million such taxpayers, and almost all of those would be corporations, not individuals. So by 2225, 5 bil doesn't seem too unrealistic.
There are still going to be all kinds of service economy jobs by then, because it would be ridiculously expensive to replace a minimum-wage earner with a robot. New colonies will need unskilled and low-skilled workers as much as the home planet - just not as many of them.
People working at a discount baby-clothes outlet aren't going to be paying galactic-scale taxes. But, if they pass the tests and volunteer for the week-long hypnotraining course in Orbital Induction Welding, suddenly they will be earning enough that their taxes matter in the galactic economy.
Yeah, that's a hell of alot of rationalization, but it works for me.