prove it!!!!!!!!!!!!!!!

Prove that:-

For any natural number n:- $n \geq 2(n!)^{\frac{1}{n}}-1$

Please post a non induction proof

#Algebra #Inequalities #Proofs #FindX #FindXandY

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Take numbers 1,2,3.....n Then $A.M \geq G.M$

Krishna Sharma - 6 years, 9 months ago

I also did in the same way

Aman Sharma - 6 years, 9 months ago

This can be re-arranged to this inequality

${ \left( 1+\frac { 1 }{ n } \right) }^{ n }\ge 2\displaystyle\prod _{ k }^{ n }{ \frac { k }{ n } }$

For $n=1$ , we have $2=2$ . Thereafter, the left side approaches $e$ while the right side drops towards $0$

Michael Mendrin - 6 years, 9 months ago

Realy intresting proof

Aman Sharma - 6 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$