Answered By Alex M

Excuse my somewhat inelegant drawing, but it's important to have a visual image of what we're working with here. Ignore some of the jargon in the question problem – who cares who Sierpinski is? And for this question, we don't really have to worry about what a fractal is or isn't. We start with a triangle, and then we divide it into four smaller triangles. It might take a few tries to see what this looks like, but I've included a basic diagram at the bottom. With this in mind, let's try to understand the first few questions.a) If the first triangle is 100 cm², calculating the shaded area of the second triangle is simple. We know that all four triangles are equal size, and three out of four are shaded. Three out of four is  $\frac{3}{4}$34   or 0.75, so multiplying that by 100 gives us 75cm².

Think about what the second stage of the question is asking. Each of the three black triangles is being divided in the same way. We could think about this in a few different ways. For example, we know that each of the small triangles is a third of the total area of 75cm², or 25cm², and multiply each by 0.75, and add up the totals. But a simpler explanation is that the total area of 75cm² is being multiplied by 0.75, even if it's happening in a slightly different way from last time. That gives us 56.25 cm².b) This should give us an idea of the process that is working here in general. We're starting with an area of 100, and multiplying by 0.75 each time. The easiest way to represent this for the nth term is 100 $\times$× 0.75^n-1. We write n-1 instead of n because the question tells us that the first triangle has an area of 100. They could have just as easily said that what they consider the first triangle was the triangle after the first division, so just be careful what's specified in the question.c) With our formula, it's easy to figure out the area at any particular point. All we have to do is plug in the right number for n. At this stage, it's just calculator work: 100 $\times$× 0.75^10-1 is 7.50846862793. Rounded to the nearest whole number, it's just 8.

d) This is a question for Math 20, so you're not expected to know calculus which would actually give you the answer directly. But we don't need to know calculus to try to think our way around the problem. We started with an area of 100. We know from the previous question that, after the tenth iteration, we're down to an area of 8. So we know that the area is decreasing. The main question for us now is whether it's decreasing towards zero or towards some other specific number. You could make a graph with the little bit of information we have from the previous questions. At n=1, it's 100, at n=2 it's 75, at n=3 it's 56.25, at n=10 it's somewhere around 7.5. We could punch a few numbers into our calculator to see what the answer is at a 15 or 20 or even 100. So the fifteenth term is 1.78179480135, the twentieth is 0.42282825852, and the hundredth is 4.2762696x10^-11. It's so small, depending on your calculator, it might show up as zero. It seems like it's going to zero! Another way we could think about this is to think about what 0.75ⁿ means, or to think of $\left(\frac{3}{4}\right)^n$(34 )ⁿ. If we expand the exponent, we can think of it as being  $\frac{3^n}{4^n}$3ⁿ4ⁿ . With a bit of logic, we can figure out that the top and bottom are getting bigger, but they're getting bigger at different speeds. This doesn't tell us that the answer is actually zero, but it helps us understand why it is. So it's a weird thing to say, but the answer, at infinity, is that the area is actually zero!