Computers in school

Andres Valloud sqrmax at prodigy.net
Tue Aug 7 17:16:49 UTC 2001


Hi.

> By "boredom" I mean a bunch of things that make a class
> un-interesting.  Assignments that have no real purpose or don't lead
> to real artifacts can lead to "boredom."  Being forced to use
> strategies that don't work for you can lead to "boredom."

Ummm... something I remember along these lines was to prove that 7^n -
2^n is divisible by 5 for all n.  The bad thing was to be forced to
prove it by induction.  That proof is horrible and unnecessarily
complicated, especially compared to using Newton's binomial theorem
directly (with the aid of the associative law for addition of integers,
and the distributive law for integers regarding multiplication and
addition).


Exercise. Prove that 7^n - 2^n is divisible by 5 for all n.

If n=0, 7^n - 2^n = 0 and there's nothing to prove since 5*0 = 0.
Otherwise, 7^n = (5+2)^n. Using Newton's binomial theorem,
	(5+2)^n = 5*(a positive amount of terms) + 2^n.
Therefore 7^n - 2^n = 5*(a positive amount of terms).
Hence it's divisible by 5, end of proof.

Andres.




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