Quote:
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Originally Posted by Pip
First trip, he goes 1 ft, and is within 2 ft of Achilles, then the band stretches 3 ft. He is then proportionally moved so that he is now 4 ft away from Achilles. He then walks another feet, now he's 3 ft from Achilles. The band stretches 3 ft, he's now 4.5 ft from Achilles, moving again... I put it up in Excel because my analytical skillz have dried away. Distance to Achilles before stretch got lower than one after travelling 10 ft, the band had then stretched to 33 feet. He only needs to go 0.781349 ft to reach Achilles then.
EDIT: Defining xn as the distance to Achilles after a stretch, and setting x0=3 (starting point) one gets
xn=(x(n-1)-1)*n/(n-1)
n will be how far the turtle has actually walked, the length of the rubber band after a stretch will be 3*(n+1).
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Yeah, that looks right. I didn't go through the entire thing, but your logic is correct as far as I can see.
To explain it in another way:
First, the turtle walks a foot. He is now 1/3 of the way across the 3 foot band. Then the band is stretched 3 feet. He is
still 1/3 (aka 2/6) of the way across, but now the band is 6 feet long.
The turtle is now 2/6 of the way across, and he moves 1 foot more. He is now 3/6 (or 1/2) of the way across. The band gets stretched 3 feet more, to 9 feet, but once again the turtle it
still 1/2 of the way across. He is 4.5/9 (Yeah, it's improper. Forget it.) of the way across.
He moves once more, and is now 5.5/9 of the way across the band. (FINE. It's 11/18. Happy?) In this manner, he does continue getting closer to the flower every step he takes. This logic is what Pip's formula describes (and what Shakran was explaning earlier).
X
n=(X
(n-1)-1)*n/(n-1)
(I didn't make that formula, I just edited Pip's to be a little more readable)
Then you just solve that for X
n=1. You'll get some decimal. You take the next whole number after that, and that's how many steps it takes the turtle to arrive at the flower. I didn't go through that, but I'm assuming Pip is right.
Edit: It's a tortoise. Oops.