wu :: forums « wu :: forums - Inequality » Welcome, Guest. Please Login or Register. Sep 7th, 2024, 9:34am RIDDLES SITE WRITE MATH! Home Help Search Members Login Register
 wu :: forums    riddles    putnam exam (pure math) (Moderators: Grimbal, Eigenray, towr, Icarus, SMQ, william wu)    Inequality « Previous topic | Next topic »
 Pages: 1 Reply Notify of replies Send Topic Print
 Author Topic: Inequality  (Read 995 times)
ThudnBlunder
Uberpuzzler

The dewdrop slides into the shining Sea

Gender:
Posts: 4489
 Inequality   « on: Aug 15th, 2006, 8:58pm » Quote Modify

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.

Prove that for all n sqrt(2sqrt(3sqrt(4sqrt(5..........sqrt(n).........)))) < 3
 « Last Edit: Aug 15th, 2006, 9:56pm by ThudnBlunder » IP Logged

THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
Eigenray
wu::riddles Moderator
Uberpuzzler

Gender:
Posts: 1948
 Re: Inequality   « Reply #1 on: Aug 15th, 2006, 11:51pm » Quote Modify

 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.
 IP Logged
srn437
Newbie

the dark lord rises again....

Posts: 1
 Re: Inequality   « Reply #2 on: Sep 23rd, 2007, 9:26pm » Quote Modify

I would disprove it, but my scientific calculator gets stack error after 13 or something around there.
 IP Logged
ima1trkpny
Senior Riddler

"Double click on 'Yes'... Hey!"

Gender:
Posts: 452
 Re: Inequality   « Reply #3 on: Sep 23rd, 2007, 9:45pm » Quote Modify

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.

LOL... update to a decent graphing/programable calculator... it will save you quite a bit of trouble.
 IP Logged

"The pessimist sees difficulty in every opportunity. The optimist sees the opportunity in every difficulty." -Churchill
srn437
Newbie

the dark lord rises again....

Posts: 1
 Re: Inequality   « Reply #4 on: Sep 23rd, 2007, 10:30pm » Quote Modify

it is decent. And also, it's not software, it's an actual calculator, so it has finite memory.
 IP Logged
mikedagr8
Uberpuzzler

A rich man is one who is content; not wealthy.

Gender:
Posts: 1105
 Re: Inequality   « Reply #5 on: Sep 23rd, 2007, 11:53pm » Quote Modify

Why not just use pen and paper?
 IP Logged

"It's not that I'm correct, it's that you're just not correct, and so; I am right." - M.P.E.
ima1trkpny
Senior Riddler

"Double click on 'Yes'... Hey!"

Gender:
Posts: 452
 Re: Inequality   « Reply #6 on: Sep 24th, 2007, 1:05am » Quote Modify

on Sep 23rd, 2007, 11:53pm, mikedagr8 wrote:
 Why not just use pen and paper?

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.)
 IP Logged

"The pessimist sees difficulty in every opportunity. The optimist sees the opportunity in every difficulty." -Churchill
mikedagr8
Uberpuzzler

A rich man is one who is content; not wealthy.

Gender:
Posts: 1105
 Re: Inequality   « Reply #7 on: Sep 24th, 2007, 1:16am » Quote Modify

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.)

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.
 IP Logged

"It's not that I'm correct, it's that you're just not correct, and so; I am right." - M.P.E.
srn437
Newbie

the dark lord rises again....

Posts: 1
 Re: Inequality   « Reply #8 on: Sep 24th, 2007, 6:23pm » Quote Modify

Pen and paper for stacking roots?! That's crazy! Anyway, square both sides until the left side is rational. 3^(2^n)>n!
 IP Logged
 Pages: 1 Reply Notify of replies Send Topic Print

 Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »