A rubber band is 1 meter long. An ant starts at one end, crawling at 1 millimeter per second. At the end of each second, the rubber band is instantaneously stretched by an additional meter. (So, at the end of the nth second, the rubber band becomes n+1 meters long.)

Does the ant ever reach the far end of the band? If so, when?

Lets assume that the ant completes the journey at time T. This means that at time T - 1 after the stretching, he had a distance d ≤ 1 mm to go. At time T - 1 before the stretching, he'd had a distance of d(T - 1)/T to go.

At time T -2 after the stretching he would have d(T - 1)/T + 1 = (dT -d +T)/T to go. Before the stretching it would have been (dT - d + T)(T - 2)/T(T -1).

I was hoping to find a pattern I could use by taking a different approach, but if there is a pattern, I can't see it. Maybe someone can pick it up.

Posted by TomM on 2002-10-03 19:43:33