Will upgrade from G to N adapter increase G reception range?

I am a university student, and the university (after I give it thousands of dollars from student loans) provides some wired internet access and some wireless. The wireless is less than ideal and without knowing what routers they are using I was wondering if a simple upgrade to a better adapter would result in better range? Just thinking about how it may work, maybe by going from G to N I gain the ability to receive N signals which travel farther but only a better antenna would increase my range? Or having MIMO? I'm not an expert. Actually im just looking into this stuff today haha

What about this "N+" from belkin? Both must be N+ to receive the advantage? I don't think the other brands even use the term "N+"