   SEARCH HOME Math Central Quandaries & 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 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     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.