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Topic: Inequalities: a problem
Replies: 10   Last Post: Jul 12, 1996 9:56 AM

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Stephen.Donnelly

Posts: 14
Registered: 12/12/04
Re: Inequalities: a problem
Posted: Jul 10, 1996 5:31 AM
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In fact I think it can be generalised to any number k
of variables:

FACT:
Let a_1 .. a_k be distinct integers.
Then \sum [a_i - a_(i+1)]^2 >= 4k - 6 .

You can prove it by induction on k now.
The case k=2 is clear.
Then if there is a counter-example to the case k,
you can easily make it into a counter-example to
the case k-1 . (By obliterating the largest integer.)

So this is really induction in the style of Fermat's
method of infinite descent.

OK?

Steve







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