The Sierpinski Triangle is a fractal, with the overall shape of an equilateral triangle.

To make a Sierpinski triangle, take an equilateral triangle, then draw three small triangles, one in each angle, then repeat it with those 3 new triangles, and repeat it again, and again...

Now, if we call the 1st step (when you have only 1 triangle) "Step 0", the next "Step 1", then "Step 2", "Step 3", and so on,

- How many triangles do we have at:
- Step 6 ?
- Step 8 ?
- Step 19 ?
- Step 50 ?

- How can you find the number of triangles drawn at Step
*n*?

I'll put a mark out of 20.

*Note: The "empty" triangles in the middle aren't included.*

EinsteinJr
May 10, 2015

#1**+5 **

**How many triangles do we have at:**

**Step 6 ?****Step 8 ?****Step 19 ?****Step 50 ?**

*I. The "empty" triangles in the middle are included:*

**$$\\ \begin{array}{rcrcrcr} s_1 &=& 1 &=& 1 &=& 1\\ s_2 &=& 3s_1+1 &=& 4 &=& 1+3\\ s_3 &=& 3s_2+1 &=& 13 &=& 1+3+3^2\\ s_4 &=& 3s_3+1 &=& 40 &=& 1+3+3^2+3^3\\ s_5 &=& 3s_4+1 &=& 121 &=& 1+3+3^2+3^3+3^4\\ s_6 &=& 3s_5+1 &=& 364 &=& 1+3+3^2+3^3+3^4+3^5\\ \cdots\\ s_n &=& 3s_{n-1}+1&=& && \dfrac{3^n-1} {2} \end{array}$$**

**$$\\ \rm{a.)}\qquad s_6 = \dfrac{3^6-1} {2} = 364 \\\\ \rm{b.)}\qquad s_8 = \dfrac{3^8-1} {2} = 9841\\\\ \rm{c.)}\qquad s_{19} = \dfrac{3^{19}-1} {2} = 581130733\\\\ \rm{d.)}\qquad s_{50} = \dfrac{3^{50}-1} {2} = 358~ 948~ 993~ 845~ 926~ 294 ~385~124$$**

* II. The "empty" triangles in the middle aren't included:*

**$$\\ \begin{array}{rcrcrcr} s_1 &=& 1 &=& 1 &=& 3^0\\ s_2 &=& 3s_1 &=& 3 &=& 3^1\\ s_3 &=& 3s_2 &=& 9 &=& 3^2\\ s_4 &=& 3s_3 &=& 27 &=& 3^3\\ s_5 &=& 3s_4 &=& 81 &=& 3^4\\ s_6 &=& 3s_5 &=& 243 &=& 3^5\\ \cdots\\ s_n &=& 3s_{n-1}&=& && 3^{n-1} \end{array}$$**

*$$\\\rm{a.)}\qquad s_6 = 3^5 = 243 \\\\\rm{b.)}\qquad s_8 = 3^7 = 2187 \\\\\rm{c.)}\qquad s_{19} = 3^{18} = 387420489 \\\\\rm{d.)}\qquad s_{50} = 3^{49} = 239 ~299 ~329 ~230 ~617 ~529 ~590 ~083$$*

heureka
May 10, 2015

#2**0 ***CONGRATULATIONS** !*

Twice as good as I expected: You've got

$$\textcolor[rgb]{1,0,0}{40/20}$$

You've earned a cookie:

EinsteinJr
May 10, 2015