I just thought of this, and I like the way that a little intuition goes a long way in setting up the simple algebra needed to solve it.

You have a starting point of 0 and an end point of 1.

Move the starting point to the middle, giving start and end points of ½ and 1.

Move the end point half way, giving a starting point of ½ and end point ¾.

Now repeat the above two steps, alternately moving the starting and end points half way. Where do you eventually end up?