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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,

  1. How many triangles do we have at:
    1. Step 6 ?
    2. Step 8 ?
    3. Step 19 ?
    4. Step 50 ?
  2. 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

Best Answer 

 #3
avatar+90996 
+5

Thanks Heureka   

 

Here is a vey interesting you tube clip on FRACTALS

https://www.youtube.com/watch?v=XwWyTts06tU

Melody  May 11, 2015
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3+0 Answers

 #1
avatar+18712 
+5

How many triangles do we have at:

  1. Step 6 ?
  2. Step 8 ?
  3. Step 19 ?
  4. 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
avatar+886 
0

Twice as good as I expected: You've got

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

CONGRATULATIONS !    

You've earned a cookie:

EinsteinJr  May 10, 2015
 #3
avatar+90996 
+5
Best Answer

Thanks Heureka   

 

Here is a vey interesting you tube clip on FRACTALS

https://www.youtube.com/watch?v=XwWyTts06tU

Melody  May 11, 2015

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