I just can't get why the **** would they make you the Archmage. You joined yesterday, you have no experience, if you just started the game, you have no magical experience at all, since the only way of acquiring good spells is buying them, literally all the other members have seniority over you, and you didn't even **** the archmage or anything. Why would they do that at all?
You know what'd be nice? Different ways to be accepted as a member based on your magic expertise. It makes no sense that a lv.30 mage, who can pretty much turn the entire hold to ash without running out of magicka, be given a set of apprentice robes and watch a class (the only class you get, by the way) about wards.
If you're an apprentice level mage, you should attend classes to learn spells instead of buying them, for instance. Or better yet, you could apprentice under one of the masters depending on the school you wanted to learn, and they would train you and teach you spells in exchange of having you doing chores, tests, errands and quests. And as you became more and more proficient in the magic arts, you could advance your post. Would have been entirely more exciting, creative and fulfilling.
Or, if you are more experienced, you could instead accomplish meaningful tasks that benefit the guild.
Ideally, once you have advanced enough, you wouldn't become Archmage and you could be instead a Master Wizard, with your own responsibilities and apprentices(they could be you followers, and you could teach them magic as you go) to booth. But since Bethesda simply must make you the single most competent person in the world, it's at least more "buy-able" that you became archmage simply because you proved to be a prodigy the likes of which is rarely seen on Tamriel or some such.