Quote:
Originally Posted by LazyScot
Spoiler:
Yes, he'll finish.
Observation 1
When the track stretches by 100m, he moves with it, so is position on the track as fraction of the total track stays the same. For example, after the first second he has travelled 10 meters, or one tenth of the track. When it expands, he is still at one tenth of the track.
Observation 2
Each second, the distance he runs, expressed as a fraction of the overall track length gets smaller. Mathematically for second n, he travels 1/ (10 *n). For example, for the first second, he travels 1/10 of the track, for the next second he travels 1/20 of the track, and so on.
Conclusion
At any time n, his position on the track is the sum of 1/10 + 1/20 + 1/30 ... + 1/(10*n).
This is equivalent to 1/10 * ( 1/1 + 1/2 + 1/3 + ... 1/n).
(1/1 + 1/2 + 1/3 + ... 1/n) is a sequence which is known to diverge, i.e. continually grow larger as n increases, and so any value will eventually be reached (google/wikipedia on Harmonic sequences).
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Very good indeed — the right answer, and a good proof that it's correct.