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Topic: Inequality (Read 1007 times) |
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ThudnBlunder
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on Aug 15th, 2006, 5:13pm, SWF wrote: By the way, we seemed to hurting for new riddles around here so I was just trying to supply one.
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Prove that for all n sqrt(2sqrt(3sqrt(4sqrt(5..........sqrt(n).........)))) < 3
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| « Last Edit: Aug 15th, 2006, 9:56pm by ThudnBlunder » |
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Eigenray
wu::riddles Moderator
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Re: Inequality
« Reply #1 on: Aug 15th, 2006, 11:51pm » |
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| hidden: | Taking log=log2, we are to show
L = log(2)/2 + log(3)/22 + log(4)/23 + log(5)/24 + log(6)/25 + ... < log 3.
Now,
log(5)/24 + log(6)/25 + ... < log(8)(1/24+...+1/27) + log(16)(1/28+...+1/215) + log(32)(1/216+...+1/231) + ...
= 3(1/24+...) + (4-3)(1/28+...) + (5-4)(1/216+...) + ...
= 3/23 + 1/27 + 1/215 + ...
< 3/8 + 1/64.
That is,
L < log(2)/2 + log(3)/4 + log(4)/8 + (3/8+1/64) = 1 + log(3)/4 + 1/8 + 1/64.
To show L < log(3), it suffices to note
24L < 243 21/221/16
< 16 3 (3/2) (1+1/9) = 80 < 81. |
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srn437
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the dark lord rises again....
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Re: Inequality
« Reply #2 on: Sep 23rd, 2007, 9:26pm » |
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I would disprove it, but my scientific calculator gets stack error after 13 or something around there.
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ima1trkpny
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Re: Inequality
« Reply #3 on: Sep 23rd, 2007, 9:45pm » |
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on Sep 23rd, 2007, 9:26pm, srn347 wrote:| I would disprove it, but my scientific calculator gets stack error after 13 or something around there. |
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LOL... update to a decent graphing/programable calculator... it will save you quite a bit of trouble.
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"The pessimist sees difficulty in every opportunity. The optimist sees the opportunity in every difficulty." -Churchill
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srn437
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the dark lord rises again....
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Re: Inequality
« Reply #4 on: Sep 23rd, 2007, 10:30pm » |
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it is decent. And also, it's not software, it's an actual calculator, so it has finite memory.
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mikedagr8
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Re: Inequality
« Reply #5 on: Sep 23rd, 2007, 11:53pm » |
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Why not just use pen and paper? 
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ima1trkpny
Senior Riddler
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Re: Inequality
« Reply #6 on: Sep 24th, 2007, 1:05am » |
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on Sep 23rd, 2007, 11:53pm, mikedagr8 wrote:| Why not just use pen and paper? |
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I would have suggested that also... but I have a strong feeling he can't do it without the help of a calculator... so I was just suggesting he use the best tool available that he is capable of understanding. And LOL, the beauty of programable graphing calculators (since he's in school) is that you can program common formulas in so all you have to do is give it the variables. (However I would recommend knowing how to do these things without a calculator first.)
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"The pessimist sees difficulty in every opportunity. The optimist sees the opportunity in every difficulty." -Churchill
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mikedagr8
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Re: Inequality
« Reply #7 on: Sep 24th, 2007, 1:16am » |
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on Sep 24th, 2007, 1:05am, ima1trkpny wrote:
I would have suggested that also... but I have a strong feeling he can't do it without the help of a calculator... so I was just suggesting he use the best tool available that he is capable of understanding. And LOL, the beauty of programable graphing calculators (since he's in school) is that you can program common formulas in so all you have to do is give it the variables. (However I would recommend knowing how to do these things without a calculator first.) |
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Hey, I have a programmable calculator. I only use it for a few equations, but these are equations which take me a full page or more of working out.
Anything I need to know I memorise, well as much as I can. And you are correct, there is no point in using something on a calculator unless you know how to do it by hand.
On tests I have lost too many marks because I was lazy and just reffered to the calculator instead of showing my working out. E.g. Simultaneuos equations.
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srn437
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the dark lord rises again....
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Re: Inequality
« Reply #8 on: Sep 24th, 2007, 6:23pm » |
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Pen and paper for stacking roots?! That's crazy! Anyway, square both sides until the left side is rational. 3^(2^n)>n!
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