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Topic: Small but powerful triangles
Replies: 1   Last Post: Apr 21, 2005 8:50 AM

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MJ

Posts: 10
Registered: 4/21/05
Re: Small but powerful triangles
Posted: Apr 21, 2005 8:50 AM
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"amcwill417" <amcwill417@email.msn.com>
wrote in message news:JPw9e.7$S23.465@eagle.america.net...

> All positive integers. Suppose (k)^n + (k+1)^n > (k+2)^n. What about an
> explicit expression (in terms of n) for the smallest value of k for which
> this is true (for any given n)?
> Alex




(k)^n + (k+1)^n > (k+2)^n.

Equivalently,

(x-1)^n + x^n > (x+1)^n

(x+1)^n - (x-1)^n < x^n

d/dx (2*x^n) < x^n

2n*x^(n-1) < x^n

x > 2n

So the answer is around 2n.

MJ





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