So, there's this game that my mom has for piano students, and a problem came up that I feel like should have a simple answer but I can't think of it. To make it simpler, I'll turn the game into this: You start at 0. You roll a die with n sides. If the number on the die is greater than the number you are at, you move to that number. If the number on the die is less than the number you are at, you win.
So, my question is, how many turns on average does it take to win, as n tends towards infinity? This game seems horribly unbalanced in that it should only take a couple turns to win, but what would this average be? It seems like the type of problem where e would sneak up on you but I can't think of any way to actually solve it.