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\[\large \left ( \dfrac {2}{3} \right ) ^ n n! \leq \left ( \dfrac {n + 1}{3} \right )^n\]

Prove the above inequation is true for all positive integers \(n\).

Note by Sharky Kesa
2 years, 5 months ago

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Apply AM -GM on \( 1,2,3...,n \).

AM = \( \dfrac{1 + 2 + 3 + ... + n}{n} = \dfrac{\frac{n(n+1)}{2}}{n} = \dfrac{(n+1)}{2} \)

GM = \( \sqrt[n]{1*2*3*...*n} = \sqrt[n]{n!} \)

Now, AM \( \geq \) GM

\( \implies \dfrac{(n+1)}{2} \geq \sqrt[n]{n!} \)

\( \implies \left (\dfrac{(n+1)}{2} \right )^n \geq n! \)

\( \implies \left ( \dfrac{(n+1)}{3} \right )^n \geq (n!) \left (\dfrac{2}{3} \right )^n \)

Siddhartha Srivastava - 2 years, 5 months ago

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Same method. @Sharky Kesa use of AM>=GM is a very common inequality in questions involving "!"

Aditya Kumar - 2 years, 4 months ago

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How about using another method?

Sharky Kesa - 2 years, 4 months ago

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By Reverse Rearrangement inequality,

\[\Large \left( \dfrac{1}{3} + \dfrac{1}{3} \right) \left( \dfrac{2}{3} + \dfrac{2}{3} \right) \ldots \left( \dfrac{n}{3} + \dfrac{n}{3} \right) \]

\[\Large \leq \left( \dfrac{1}{3} + \dfrac{n}{3} \right) \left( \dfrac{2}{3} + \dfrac{n-1}{3} \right) \ldots \left( \dfrac{n}{3} + \dfrac{1}{3} \right)\]

So, on simplification the above inequality reduces to

\[\Large \left( \dfrac{2}{3} \right) ^{n} n! \leq \left( \dfrac{n+1}{3} \right) ^ {n} \]

which is our desired result.

Surya Prakash - 2 years, 3 months ago

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Nice :). Sorry nowadays I can't come on hangouts. Tell other guys also.

Aditya Kumar - 2 years, 3 months ago

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Yup! I will say. :D

Surya Prakash - 2 years, 3 months ago

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Yeah I thought about Daniel's Reverse Rearrangement inequality.

Chinmay Sangawadekar - 2 years, 4 months ago

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