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

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

thnkx in advnc.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,654

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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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,057

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

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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,654

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

simply amazing!:)

thank you all

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