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| Math Central | Quandaries & Queries |
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Question from Vicki, a student: I am trying to find out how to do show how this proof was worked. This equation was used to find the number of white triangles in the Sierpinski Triangle |
Vicki,
It seems that an induction is not the most transparent reasoning here.
Here is a pattern which does not depend on n:
\[1 + 3 + 3^2 ...+3^n-1 = S\]
Multiply by 3:
\[3 + 3^2 ....+ 3^n-1 + 3^n = 3S\]
Subtract the first from the second:
\[-1 + 0+0+ ... +0 + 3^n = 2S\]
Divide by 2 and you have your solution.
Using induction is possible, but does not add to the understanding of most students.
At a key level, I believe in Street Fighting mathematics:
http://ocw.mit.edu/courses/mathematics/18-098-street-fighting-mathematics-january-iap-2008/
Do what works, has meaning, solves the problem .... .
Walter Whiteley
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