This is a nice problem! This is from high school???
Let's denote given cards, let be the number on the first card you pick, and let be the random variable of the number of cards you pick for cards. The reason, I have all the different random variables is that we are going to end up with a recurrence relation.
Ok, here is the trick. Suppose you pull a 4 (X=4). From the standpoint of , you could just throw away the 1, 2, and 3 cards from the deck. They do not impact anymore. So now you have a deck containing 5 to N. The position you are in now is exactly the same as if you had N-5 cards and you hadn't started yet except that you already picked a card. So you get:
A little manipulation and you get:
This is a nasty recurrence relation so let's try to get rid of most of the terms. So then:
Now multiply (2) by and subtract from (1) (so we can eliminate most of the terms in the sum).