00:01
So in this question, we're thinking about the koch snowflake, which is a fractal where we start off with a triangle, and then we add a triangle to each side, and we continue adding a triangle to each side forever.
00:24
So how many triangles are we adding at the end step? so this is the first step.
00:27
This is the second step.
00:30
So at the first step, we add three triangles.
00:35
At the second step, we add 12 triangles.
00:37
And what we're doing after that is that for each triangle, we add a triangle next to it, two triangles on it, and a triangle on the other side.
00:47
So for each triangle we added in the previous step, so say at the nth step, we add some n -n -n, then at the n -plus -1th step, we add four times as many triangles as we did last time, 4 -n -n.
01:06
So that means that n n plus 1 equals 4 times n n.
01:13
So the number that we're adding is going to be 4 to the power of n.
01:25
So this is going to be, so the number that we're adding is going to be 4 to some power.
01:34
So we add 3, so it's 4.
01:39
So it's three times four to the power of this to the minus one.
01:45
So it's going to be three times four to the power of n minus one.
01:53
Because that way, at each step, we're multiplying them by a factor of four.
01:59
And actually, on each side, it's just we add one in the first step, so that's four to the power of zero.
02:07
We have four in the second step, so that's four to the power of one.
02:10
But there's three sides.
02:13
So that's where the three comes from, and then the four to the power of n minus one.
02:17
That's how many we add at the end step.
02:22
So at each step, what's the area of the new triangle? so at the first step, so let's say that the first triangle has an area of a.
02:33
Then at the next step, the length of the sides is a third of what it was before, which means that a1 is going to be a third square, times a, so that's a over 9...