Math CentralQuandaries & Queries


Question from Vicki, a student:

I am trying to find out how to do show how this proof was worked.
Here is the end result 1 + 3 + 3^2 ...+3^(n-1) = 3^n - 1/2

This equation was used to find the number of white triangles in the Sierpinski Triangle


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:
Do what works, has meaning, solves the problem .... .

Walter Whiteley

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