So, the answer to that revised question is: It depends. The national recommendations are 150 minutes of low to moderate intensity or 75 minutes of high intensity exercise (or a combo of both) per week to maintain
current health and fitness. Read that again: MAINTAIN current health and fitness. Those numbers are the minimum requirements--meaning you won't improve your health and fitness unless you do more than that. Now, if someone is sedentary and just coming off the couch of course they will improve for a time, but soon that will level out to where their fitness level should be and they will need to do more.
I know, being of rational mind you're seeing '150' and '75' and thinking those numbers are to clean and rounded and can't possibly be scientifically accurate, and I would agree. The real numbers are different for everyone and without expensive equipment loaded with ding dongs, wires, and whosiewhatsits, there's no accurate way to pinpoint the numbers for each individual. They are really meant to be used as guidelines from where we can adjust per individual, and those guidelines have been relatively successful so far. It's the same as saying 3500 calories equals a pound. From the research I've read, it's really closer to 3554, but since we simple humans like things in nice, tidy packages, 3500 it is.
Back to the topic at hand.
The number of sessions you need to do per week depends on what you do when you're not with your trainer. If you see your trainer once a week for 60 minutes of high intensity exercise and don't do anything the next six days, you're not even meeting your maintenance goal. If one session per week is all you've budgeted for, then you need to make up the remainder of the maintenance minutes, plus the extra for improvement, on your own.
Let's say you buy 40 sessions at 60 minutes each and you plan on those 40 sessions being your only source of exercise until they are exhausted. Scheduling 2 per week is going to have vastly different results than if you scheduled those same 40 sessions at 3-4 per week. At 2 per week, you're only getting 120 minutes, where 3-4 per week gets you to your maintenance goals plus the extra needed to improve. On the flip-side, if you're overtraining, a number of reasons will keep you from improving--torn fibers that need to rest and repair, tiredness and soreness causing you to be not as effective, etc.
So really, no matter how many sessions you order, just make sure you are scheduling them close enough together (keeping in mind rest days) to make a difference, otherwise you're wasting your time and money.