MiniProiect#3-Fractals y purpose for this project is for you to apply Yooe for c

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MiniProiect#3-Fractals y purpose for this project is for you to apply Yooe for completing series to another branch of communicate to me your is a formal ematical study. Your purpose for completing this project is to rstanding of geometric sequences as they apply to fractals. This your audience and you are expected to write in the genre of apply your knowledge of sequences and assignment where I am audience handwritten hufo incoting formulas and equations as necessary. The project st fanother sheet of paper with answers clearly marked. Answers should be communicate in written on is that you will use proper English and complete sentences to should be done on given in exact form. ih out all work must be included and shown in an organized manner. All work sheet Fractal: A fractal is a geome tric shape that can be split in parts that are reduced scale patterns part of the original. This pattern is repeated an infinite number of it you will see it looks the same. e Sierpinski Triangle. This fractal is created by connecting the midpoints of ee sides of an equilateral triangle. This creates a new equilateral triangle which is then removed" from the original. The remaining three triangles are smaller versions of our original. One such fractal is the process can then be repeated to continue to create other iterations of the figure. The first three stages are shown here: Stage 2 Stage 3 Stage 1 Part I: a) Write a sequence to represent how many shaded triangles there are at each stage. b) Based on this sequence, write a formula for the nth term of the sequence. c) Using at least two sentences, describe what is represented by each term in the sequence.

Explanation / Answer

stage 1 --> 1
stage 2 ---> 3
stage 3 ---> 9
stage 4 ---> 27
etc etc etc

So, the sequence is :
1 , 3 , 9 , 27 , 81 , .....

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b)
Clearly this is of the form of multiple of 3

So, an = 3^(n-1) ---> ANS

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c)
The 3 is the common ratio
and the n is the term number
starting with first term n = 1

So, if want to find the 10th term,
i.e the number of shaded triangles in the 10th stage, we simply do 3^(10-1) --> 3^9 to get the answer

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part II :
a)
at nth stage, the side length is (a) / 2^(n-1)
and height is : a*sqrt(3)/2 * 1/2^(n-1)
and area is : sqrt(3) * a^2 * 1/4^n

b) ht of stage 1 is a*sqrt(3)/2

c) Area = sqrt(3)/4 * a^2

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part III :
a) a*sqrt(3)/2 * 1/2
= asqrt(3)/4

b)
Area = sqrt(3) * a^2 * 1/4^n
sqrt(3) * a^2 * 1/16
sqrt(3)/16 * a^2

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part IV :
a)
Let the original length of the first triangle be a
Clearly stage 1 ----> a
Stage 2 ---> a/2
Stage 3 ---> a/4
Stage 4 ---> a/8

And thus,
at nth stage, the side length is (a) / 2^(n-1)

b)
Clearly for a side length of a
ht is a*sqrt(3)/2

Stage 1 ---> a*sqrt(3)/2
Stage 2 ---> 1/2 * a*sqrt(3)/2
Stage 3 ---> 1/4 * a*sqrt(3)/2

So, at nth stage,
height is : a*sqrt(3)/2 * 1/2^(n-1)

c)
Area = sqrt(3)/4 * side^2

sqrt3/4 * (a/2^(n-1))^2

sqrt(3)/4 * a^2 / 4^(n-1)

So, we have
sqrt(3) * a^2 * 1/4^n

sqrt(3)*a^2 / 4^n ----> ANS

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