Sierpinski's Triangle - 2

Geometry Level 2

This problem is part of a set of problems, please see the linked first problem , if you haven't yet.

Question: How many white triangles are there at step $n$ of Sierpinski's Triangle?

Hints:

How many white triangles are created from each red triangle of the previous step?
Once a white triangle is made, it is not changed anymore.

Note: This problem is part of a set of problems, further investigating Sierpinski's Triangle in different ways. If you want to continue investigating click here . Questions will increase in difficulty.

See the next problem .

Image Credits: Sierpinski's Triangle Image created by William Andrus. Removed grey outline.

3^{n - 1}

\frac{3^{n}-1}{2}

n^{2}

3^{n}

None of these.

3^{n + 1}

1 solution

Mihael Keehi
Mar 13, 2018

At step 0 there is only one red triangle, thus there are $0$ white triangles .

At step 1 the previous red triangle is divided into four identical triangles, $1$ of which is white. Thus, there is $1$ white triangles .

At step 2 each red triangle in the previous step is divided into four identical triangles, $1$ of which is white. Therefore, since there were $3$ red triangles in the previous step, we get $3$ more white triangles. Adding the white triangle we already had, there are $1+3=4$ white triangles .

At step 3 for each red triangle in the previous step we get $1$ white. Therefore, since there are $3^2$ red triangles, we get $3^2$ more white triangles. Adding the white triangle we already had, there are $1+3+3^2=4$ white triangles .

Repeating this process, at step $n$ we have $1+3+3^2+\ldots+ 3^{n-1}=\dfrac{1-3^n}{1-3}=\dfrac{3^n-1}{2}.$ white triangles .

Sierpinski's Triangle - 2

1 solution

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