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#1 2005-07-25 17:10:40

kylekatarn
Member
Registered: 2005-07-24
Posts: 445

inequalities/proving

can someone proove (or disproof) that:
(a+b)^(n*a)>(a^2)(n^b)
:|

conditions:
a>0
b>0
n>=1

any help would be great

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#2 2005-07-25 19:32:47

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

Re: inequalities/proving

n = a natural number?

Perhaps we could start off with a proof for n=1, then onto 2 or perhaps n+1

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#3 2005-07-25 19:33:58

ganesh
Registered: 2005-06-28
Posts: 25,928

Re: inequalities/proving

The binomial expansion of the Left Hand Side
contains two terms, among many, which are
a^a*n and b^a*n which appear to make the LHS greater than the RHS,
but when we assign arbitrary values,
say a=10, b=1,000,000,000 and n=100
the LHS is (1,000,000,010)^1000, which would contain 9,001 digits;
the RHS becomes
100 x (100^1,000,000,000) which would contain more than 2 billion digits!
This happened because we assumed b>>n.
Otherwise, the LHS may be greater.
Say, when a=10, b=100, n=1000.
LHS would be 110^10,000 containing 20,414 digits and the RHS would be much smaller, viz. 100*(1000^100), containing approximately 300 digits!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2005-07-25 19:42:17

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: inequalities/proving

If you take n=1, then you can disprove both by counter-example.

Take a=1.2, b=0.00001, n=1
That gives (1.2+0.00001)^1.2*1</>(1.2^2)*1^0.00001, meaning 1.244...</>1.44.
So, in this case, a²*n^b is larger.

Almost all other cases result in (a+b)^na being larger.

e.g. (3+3)^3*1</>(3^2)*1^3, meaning 216</>9.

So, it can't be proven or disproven.

Why did the vector cross the road?
It wanted to be normal.

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#5 2005-07-25 19:46:42

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

Re: inequalities/proving

Magic, guys. Disproven by example.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#6 2005-07-26 01:06:08

kylekatarn
Member
Registered: 2005-07-24
Posts: 445

Re: inequalities/proving

simply amazing!:)
thank you all

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