Camp exam problem 4

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*Mahi*
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Camp exam problem 4

Unread post by *Mahi* » Fri Nov 04, 2011 11:32 pm

Prove that,\[\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{2n-1}{2n} \leq \frac{1}{\sqrt{3n+1}}\]
for all positive integers $n$ .
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nafistiham
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Re: Camp exam problem 4

Unread post by nafistiham » Sat Nov 05, 2011 12:49 am

simple induction
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\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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sm.joty
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Re: Camp exam problem 4

Unread post by sm.joty » Sat Nov 05, 2011 11:12 am

I'm not clear about using induction in inequality. :( So can anyone post the solution.
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

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Re: Camp exam problem 4

Unread post by sm.joty » Sat Nov 05, 2011 6:39 pm

আরে এটা দেখি "Art & Craft Of Problem Solving" থেকে আসছে। যাইহোক আগে থেকে না দেখায় ভালই হইসে...(অন্তত rules ভঙ্গ হয়নাই।) :lol:
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

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nafistiham
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Re: Camp exam problem 4

Unread post by nafistiham » Sat Nov 05, 2011 6:41 pm

last step :
\[\frac{1}{2} \cdot \frac{3}{4}\cdots \frac{2m-1}{2m} \cdot \frac{2m+1}{2m+2}\]
\[\geq \frac{1}{\sqrt{3m+1}} \cdot \frac{2m+1}{2m+2}\]
\[=\frac{1}{\sqrt{3m+4}}\]
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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