+870 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,
I'll put a mark out of 20.
Note: The "empty" triangles in the middle aren't included.
+26367 How many triangles do we have at:
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:
$$\\
\begin{array}{rcrcrcr}
s_1 &=& 1 &=& 1 &=& 3^0\\
s_2 &=& 3s_1 &=& 3 &=& 3^1\\
s_3 &=& 3s_2 &=& 9 &=& 3^2\\
s_4 &=& 3s_3 &=& 27 &=& 3^3\\
s_5 &=& 3s_4 &=& 81 &=& 3^4\\
s_6 &=& 3s_5 &=& 243 &=& 3^5\\
\cdots\\
s_n &=& 3s_{n-1}&=& && 3^{n-1}
\end{array}$$
$$\\\rm{a.)}\qquad s_6 = 3^5 = 243
\\\\\rm{b.)}\qquad s_8 = 3^7 = 2187
\\\\\rm{c.)}\qquad s_{19} = 3^{18} = 387420489
\\\\\rm{d.)}\qquad s_{50} = 3^{49} = 239 ~299 ~329 ~230 ~617 ~529 ~590 ~083$$
+870 Twice as good as I expected: You've got
$$\textcolor[rgb]{1,0,0}{40/20}$$
You've earned a cookie:
