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Zeno's Paradox: Solution

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Merlin11188 is not online. Merlin11188
Joined: 20 Dec 2007
Total Posts: 4158
26 Aug 2011 06:07 PM
Actually, my geometry teacher showed this to us, and I thought it was pretty cool (it is a geometric sequence). Before I tell you what he showed us, you must first understand Zeno's paradox:
You can never get to where you're going because to get there, you have to go halfway there (1/2), halfway there again (1/2 + 1/4 = 3/4), halfway there again (7/8), and so on, never actually reaching your destination. Alright, so here's what the solution is:
If it takes you 4 seconds to get halfway there (1/2), then it takes you 2 seconds to get another halfway there (1/4), and so on. So, since 4, 2, 1, 1/2 ... is a geometric sequence with a difference/quotient/rate (or whatever you call it) of between -1 and 1 (in this case it's 1/2), this is true:
S = t(1)/(1-r)
r is the rate, S is the sum (in this case of time), and t(1) is the first term in the sequence, which is 4.
So:
4/(1-(1/2)) = 8/1 = 8
So, the time it takes you to get to your destination is 8 seconds because 4 + 2 + 1 + 1/2 ... adds up to 7.9(repeating).

Sorry if I messed up explaining it to you; this was much harder to write down than to picture in my head...
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eyeontheprizeREBOOT is not online. eyeontheprizeREBOOT
Joined: 30 Jan 2011
Total Posts: 1555
26 Aug 2011 06:08 PM
Except the part where you rounded.
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Merlin11188 is not online. Merlin11188
Joined: 20 Dec 2007
Total Posts: 4158
26 Aug 2011 06:13 PM
So, are you saying I'm incorrect or "Zeno's Paradox" isn't really a paradox?
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eyeontheprizeREBOOT is not online. eyeontheprizeREBOOT
Joined: 30 Jan 2011
Total Posts: 1555
26 Aug 2011 06:15 PM
I'm saying that if you truly only move halfway there t will approach but never reach zero. Rounding implies you didn't truly move halfway.
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Merlin11188 is not online. Merlin11188
Joined: 20 Dec 2007
Total Posts: 4158
26 Aug 2011 06:17 PM
Alright, then, it takes you 7.9 repeating seconds to get there.
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ArceusInator is not online. ArceusInator
Joined: 10 Oct 2009
Total Posts: 30553
26 Aug 2011 06:18 PM
Exactly.
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Fungalmungal is not online. Fungalmungal
Joined: 18 Oct 2008
Total Posts: 2519
26 Aug 2011 06:35 PM
7.9 repeating is equal to 8. Zeno's paradox was never a paradox. The ancient greeks just didn't have the tools they needed for such problems, namely the concept of a limit.
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