nice and easy

VITEEE Coaching › Olympiad Stuff questions › nice and easy

show that
n! < {(n+1)/2}^2 for n belonging to N

#Mathematics #Olympiad Stuff

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7 Answers

11

Devil ·2009-08-12 10:34:57

I think this is what u intended....
n!<{n+12}^n....

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106

Asish Mahapatra ·2009-08-13 01:44:21

If that is the case
then:
n!=(1*n)(2*(n-1))*.....

(1+n)²/4 â‰¥ 1*n
(2+n-1)²/4 â‰¥ 2*(n-1)
.
.
.
continuing and multiplying
(n+12)^2(n/2) â‰¥ n!
or n! â‰¤ (n+12)ⁿ

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rahul1993 Duggal ·2009-08-13 02:34:22

@asish correct solution.
but equality is not valid.
keep trying there's a simpler solution

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rahul1993 Duggal ·2009-08-13 02:36:14

oh i forgot to mention n>1 asish your solution is perfect

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106

Asish Mahapatra ·2009-08-13 02:36:29

yeah equality not valid bcz nos are not equal... :)

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11

Devil ·2009-08-13 08:34:23

What abt AM-GM on 1st 'n' naturals?

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rahul1993 Duggal ·2009-08-13 08:41:05

that's right soumik :D

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