2003 AIME II problem 4 (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.910
In a regular
tetrahedron-- and that's

00:00:02.910 --> 00:00:05.950
just a four-sided
polyhedron, and its regular,

00:00:05.950 --> 00:00:09.460
so all of the sides and all of
the faces will be the same--

00:00:09.460 --> 00:00:12.810
the centers of the four
faces are the vertices

00:00:12.810 --> 00:00:15.140
of a smaller tetrahedron.

00:00:15.140 --> 00:00:18.770
The ratio of the volume of the
smaller tetrahedron to that

00:00:18.770 --> 00:00:22.340
of the larger is m over
n, where m and n are

00:00:22.340 --> 00:00:25.110
relatively prime
positive integers.

00:00:25.110 --> 00:00:27.130
Find m plus n.

00:00:27.130 --> 00:00:30.780
So let's just draw this
regular tetrahedron first.

00:00:30.780 --> 00:00:33.820
So this is a
regular tetrahedron.

00:00:33.820 --> 00:00:38.120
So in order to have four equal
sides, or four equals faces,

00:00:38.120 --> 00:00:39.910
I should say, in
three dimensions,

00:00:39.910 --> 00:00:43.730
it needs to look like this.

00:00:43.730 --> 00:00:47.420
It essentially needs
to be four triangles.

00:00:47.420 --> 00:00:51.080
So we have one triangle,
just like that.

00:00:51.080 --> 00:00:53.640
Then we have another
triangle like that.

00:00:53.640 --> 00:00:56.550
Then we have another
triangle in the back.

00:00:56.550 --> 00:00:58.400
So this is another
triangle in the back.

00:00:58.400 --> 00:01:00.350
And then we have a
triangle as a base.

00:01:00.350 --> 00:01:04.090
This is a regular tetrahedron,
where each of these faces

00:01:04.090 --> 00:01:07.040
are going to be
equilateral triangles.

00:01:07.040 --> 00:01:10.280
Each of these faces are
equilateral triangles.

00:01:10.280 --> 00:01:13.270
So if you looked, this length
is the same as that length,

00:01:13.270 --> 00:01:15.700
this is the same as that length.

00:01:15.700 --> 00:01:16.380
Fair enough.

00:01:16.380 --> 00:01:18.620
So that's our
regular tetrahedron.

00:01:18.620 --> 00:01:21.032
Now, they say the centers
of the four faces,

00:01:21.032 --> 00:01:22.990
so let's look at the
centers of the four faces.

00:01:22.990 --> 00:01:25.970
Let's say that this is
the center of one face.

00:01:25.970 --> 00:01:28.550
This is the center
of another face.

00:01:28.550 --> 00:01:31.380
Back here is the
center of the face

00:01:31.380 --> 00:01:34.550
that we can't see that's
behind these two faces.

00:01:34.550 --> 00:01:37.570
And then this is the center,
this right here, let's say,

00:01:37.570 --> 00:01:40.030
is the center of the base.

00:01:40.030 --> 00:01:43.470
The center of the four faces
are the vertices of a smaller

00:01:43.470 --> 00:01:45.910
tetrahedron, so we
could connect them

00:01:45.910 --> 00:01:47.760
and get a smaller tetrahedron.

00:01:47.760 --> 00:01:50.450
So along, so we
have this one going

00:01:50.450 --> 00:01:52.930
from the center of this face
to the center of the base.

00:01:52.930 --> 00:01:54.570
Going from the
center of the base

00:01:54.570 --> 00:01:57.510
to the center of this face
over here, we have this line.

00:01:57.510 --> 00:02:00.360
You connect those, we have one
of the faces of the smaller

00:02:00.360 --> 00:02:01.460
tetrahedron.

00:02:01.460 --> 00:02:04.706
You go like that, you
have kind of the top.

00:02:04.706 --> 00:02:06.580
If you flipped it over,
it would be the base,

00:02:06.580 --> 00:02:09.297
but you'd have the top of
the smaller tetrahedron.

00:02:09.297 --> 00:02:11.130
And then if you connected
those in the back,

00:02:11.130 --> 00:02:12.671
you would have the
two faces that you

00:02:12.671 --> 00:02:15.540
can't see-- that
face and that face.

00:02:15.540 --> 00:02:17.390
Now, they want us to
essentially figure out

00:02:17.390 --> 00:02:20.580
the volume of the smaller
tetrahedron, this magenta

00:02:20.580 --> 00:02:24.685
tetrahedron, to this
yellow one over here.

00:02:24.685 --> 00:02:26.560
And you might already
be wrecking your brain,

00:02:26.560 --> 00:02:30.024
how do you figure out the volume
of a tetrahedron and whatnot,

00:02:30.024 --> 00:02:30.690
and all of that.

00:02:30.690 --> 00:02:34.215
But the important
thing to realize

00:02:34.215 --> 00:02:40.560
if you take two similar
objects and if you just

00:02:40.560 --> 00:02:43.032
know the ratio of one
of their dimensions,

00:02:43.032 --> 00:02:44.740
and assuming that all
of their dimensions

00:02:44.740 --> 00:02:47.550
have the same ratio-- and
it is the case in this right

00:02:47.550 --> 00:02:50.620
over here-- then
the volume will be

00:02:50.620 --> 00:02:54.400
the cube of the ratios
of one of their sides.

00:02:54.400 --> 00:02:58.000
So let me show this to
you right over here.

00:02:58.000 --> 00:02:58.720
Let me depict it.

00:02:58.720 --> 00:03:05.270
So let's say that this
length right over here,

00:03:05.270 --> 00:03:09.710
let's call that Length Big
for the big tetrahedron.

00:03:09.710 --> 00:03:14.290
And let's find the corresponding
edge of the small tetrahedron.

00:03:14.290 --> 00:03:20.100
So let's say that
we have another side

00:03:20.100 --> 00:03:25.390
of the small tetrahedron,
let's call that Length Small.

00:03:25.390 --> 00:03:35.360
If we know the ratio of
Length Big to Length Small,

00:03:35.360 --> 00:03:42.230
we can figure out the ratio
of the big tetrahedron

00:03:42.230 --> 00:03:47.630
to the small tetrahedron, the
volume of the big tetrahedron

00:03:47.630 --> 00:03:50.780
to the volume of the
small tetrahedron

00:03:50.780 --> 00:03:52.600
by just cubing this ratio.

00:03:56.220 --> 00:03:58.670
And we can do that because
these are completely

00:03:58.670 --> 00:04:03.200
similar in every direction,
in every dimension.

00:04:03.200 --> 00:04:05.820
Whatever the ratio
of this is to this

00:04:05.820 --> 00:04:07.870
is the same as the ratio
of that is to that.

00:04:07.870 --> 00:04:10.180
And actually, all of
these sides are the same,

00:04:10.180 --> 00:04:12.510
so all of these sides are
going to be the same as well.

00:04:12.510 --> 00:04:14.620
So they're similar
in every dimension.

00:04:14.620 --> 00:04:17.079
So with that said,
we really just need

00:04:17.079 --> 00:04:21.500
to figure out the ratio
of one of the sides

00:04:21.500 --> 00:04:24.070
of the big tetrahedron
to the length

00:04:24.070 --> 00:04:26.590
of the side of a
small tetrahedron.

00:04:26.590 --> 00:04:28.340
And to do that, let's
just put ourselves--

00:04:28.340 --> 00:04:30.290
And there are more
elegant ways of doing it,

00:04:30.290 --> 00:04:32.594
but those get a little
bit more abstract.

00:04:32.594 --> 00:04:34.510
What I'm going to do is
just try to figure out

00:04:34.510 --> 00:04:37.800
some coordinates of some of
these vertices of the larger

00:04:37.800 --> 00:04:38.677
tetrahedron.

00:04:38.677 --> 00:04:41.010
And from that we can take the
average of the coordinates

00:04:41.010 --> 00:04:44.730
to find the coordinates of
these vertices, the centres

00:04:44.730 --> 00:04:46.240
of the faces.

00:04:46.240 --> 00:04:50.060
So let me draw some
coordinate axes here.

00:04:50.060 --> 00:04:53.000
So let's say that
this is my y-axis.

00:04:53.000 --> 00:04:55.710
I'm going to try to draw it
so that we can visualize it

00:04:55.710 --> 00:04:56.785
in three dimensions.

00:04:56.785 --> 00:04:58.076
So let's say that is my y-axis.

00:05:01.470 --> 00:05:05.590
Let's make my x-axis
do something like this.

00:05:05.590 --> 00:05:09.152
So let's say that is the x-axis.

00:05:09.152 --> 00:05:11.360
And then you're going to
have your z-axis go straight

00:05:11.360 --> 00:05:12.150
up and down.

00:05:12.150 --> 00:05:13.650
And actually, I
won't draw that yet.

00:05:13.650 --> 00:05:15.066
What I want to
just draw right now

00:05:15.066 --> 00:05:18.949
is the base of the
larger tetrahedron.

00:05:18.949 --> 00:05:19.990
And I'm going to do that.

00:05:19.990 --> 00:05:22.073
I'm just going to pick
arbitrary coordinates here.

00:05:22.073 --> 00:05:24.580
We really just care
about the ratios,

00:05:24.580 --> 00:05:28.420
so we can pick the larger
tetrahedron and have any size.

00:05:28.420 --> 00:05:32.470
So let's say that that
vertex right over here

00:05:32.470 --> 00:05:35.890
is sitting right over here.

00:05:35.890 --> 00:05:39.250
This vertex sits
right over here.

00:05:39.250 --> 00:05:44.290
And then this vertex right
over here sit right over there.

00:05:44.290 --> 00:05:46.310
And then let's
assign-- we'll make it

00:05:46.310 --> 00:05:50.660
so that looks a
little bit better.

00:05:50.660 --> 00:05:54.060
And so this is the base.

00:05:54.060 --> 00:05:56.540
of our tetrahedron,
of our big one.

00:05:56.540 --> 00:06:00.310
It's an equilateral triangle--
all the sides are the same.

00:06:00.310 --> 00:06:06.900
And just for simplicity
of the coordinates,

00:06:06.900 --> 00:06:09.030
let's make this coordinate
right over here,

00:06:09.030 --> 00:06:13.090
let's say that x is equal to
1, y is clearly equal to 0,

00:06:13.090 --> 00:06:15.520
because we are on the
x-axis, we haven't

00:06:15.520 --> 00:06:17.120
moved in the y direction.

00:06:17.120 --> 00:06:19.170
And then we haven't even
paid any attention to z,

00:06:19.170 --> 00:06:21.750
but it's clearly
sitting in the xy-plane,

00:06:21.750 --> 00:06:26.690
so z is also going
to be equal to 0.

00:06:26.690 --> 00:06:29.900
If we go over here, let's
make this negative 1, comma 0,

00:06:29.900 --> 00:06:30.950
comma 0.

00:06:30.950 --> 00:06:33.280
So it's still on the x-axis.

00:06:33.280 --> 00:06:36.410
So this distance right over here
is 1, this distance over here

00:06:36.410 --> 00:06:39.280
is 1, this entire side is 2.

00:06:39.280 --> 00:06:44.440
So we also know that this
entire side over here

00:06:44.440 --> 00:06:47.270
is going to be 2.

00:06:47.270 --> 00:06:49.750
And you could use
30-60-90 triangles

00:06:49.750 --> 00:06:51.300
to figure out this
distance, which

00:06:51.300 --> 00:06:53.440
we need to figure out
the y-coordinate here.

00:06:53.440 --> 00:06:54.780
Let me be very clear.

00:06:54.780 --> 00:06:56.830
This, the coordinate of
this point over here,

00:06:56.830 --> 00:06:59.250
x is clearly equal
to 0 over here.

00:06:59.250 --> 00:07:00.590
X is 0.

00:07:00.590 --> 00:07:04.930
z is clearly equal to 0,
we're still in the xy-plane.

00:07:04.930 --> 00:07:07.020
But we don't know what
the y-coordinate is.

00:07:07.020 --> 00:07:09.330
So we're going to have to
do a little bit of-- well,

00:07:09.330 --> 00:07:12.190
you could use your knowledge
of 30-60-90 triangles

00:07:12.190 --> 00:07:15.270
or if you don't even want to do
that, you can just go straight

00:07:15.270 --> 00:07:17.290
to the Pythagorean theorem.

00:07:17.290 --> 00:07:19.520
We're looking for the
y-coordinate over here.

00:07:19.520 --> 00:07:23.190
We're looking for this
distance right over here.

00:07:23.190 --> 00:07:27.380
So this y-coordinate
squared plus 1

00:07:27.380 --> 00:07:29.570
squared is going
to equal 2 squared.

00:07:29.570 --> 00:07:31.530
So let me write that over here.

00:07:31.530 --> 00:07:33.170
So this-- I will
do a new color--

00:07:33.170 --> 00:07:38.370
so the y-coordinate squared plus
1 squared-- so plus 1 is going

00:07:38.370 --> 00:07:42.350
to be equal to 2 squared,
it's going to be equal to 4,

00:07:42.350 --> 00:07:44.050
where this is the y-coordinate.

00:07:44.050 --> 00:07:46.980
So y squared is going
to be equal to 3.

00:07:46.980 --> 00:07:49.540
y is going to be equal
to the square root of 3.

00:07:49.540 --> 00:07:52.880
So this distance right over
here is the square root of 3.

00:07:52.880 --> 00:07:54.607
Or so I could write
square root of 3

00:07:54.607 --> 00:07:56.940
here-- you could see that
this satisfies the Pythagorean

00:07:56.940 --> 00:07:59.560
theorem-- or I could
just put its coordinate--

00:07:59.560 --> 00:08:03.450
it's coordinate is square
root of 3 in the y-direction.

00:08:03.450 --> 00:08:06.510
Now, while we're
focusing on the base

00:08:06.510 --> 00:08:09.360
on the base of the
large tetrahedron,

00:08:09.360 --> 00:08:12.850
we can right now figure out
the center of that base,

00:08:12.850 --> 00:08:15.540
figure out the
center of that face,

00:08:15.540 --> 00:08:18.750
which maybe might
sit right about here.

00:08:18.750 --> 00:08:21.320
And the easy way to figure
out the center of this face

00:08:21.320 --> 00:08:24.640
is literally just to
average of the coordinates.

00:08:24.640 --> 00:08:27.990
So the coordinate for this
character right over here is

00:08:27.990 --> 00:08:30.320
going to be the
average of the x's.

00:08:30.320 --> 00:08:35.580
So 0 plus 1 minus
1 is 0 divided by 3

00:08:35.580 --> 00:08:38.010
is just going to be 0,
which is very clear.

00:08:38.010 --> 00:08:41.055
It's sitting on the y-axis.

00:08:41.055 --> 00:08:43.460
It does not move in
the x x-direction,

00:08:43.460 --> 00:08:45.860
either to the right
or to the left.

00:08:45.860 --> 00:08:48.920
The y-coordinate, we're going
to have square root of 3

00:08:48.920 --> 00:08:52.760
plus 0 plus 0 over 3.

00:08:52.760 --> 00:08:56.210
So that's just going to be
square root of 3 over 3.

00:08:56.210 --> 00:09:00.080
So it's one third of the way all
the way to the top of the base.

00:09:00.080 --> 00:09:01.720
And then the
z-coordinate coordinate,

00:09:01.720 --> 00:09:03.553
you could take the
average, but it's clearly

00:09:03.553 --> 00:09:05.360
still sitting on the xy-plane.

00:09:05.360 --> 00:09:07.310
So it's also going to be 0.

00:09:07.310 --> 00:09:11.070
You can take 0 plus 0 plus 0
divided by 3-- clearly still 0.

00:09:11.070 --> 00:09:15.930
So we figured out also one of
the vertices of our smaller

00:09:15.930 --> 00:09:16.890
tetrahedron.

00:09:16.890 --> 00:09:20.460
What I want to do now
to try to figure out

00:09:20.460 --> 00:09:23.340
the coordinates of
this vertex right here.

00:09:23.340 --> 00:09:25.640
Then we can figure
out this distance,

00:09:25.640 --> 00:09:28.870
compare it to a distance of
2, and then, essentially,

00:09:28.870 --> 00:09:32.542
take the cube of it, and
we'll have the ratio we need.

00:09:32.542 --> 00:09:34.000
And then we can
figure out the rest

00:09:34.000 --> 00:09:35.880
of the problem-- the m plus n.

00:09:35.880 --> 00:09:38.770
Now, to figure out
the coordinates here,

00:09:38.770 --> 00:09:40.320
we need to take
the average of this

00:09:40.320 --> 00:09:43.260
coordinate, this coordinate,
and this coordinate up here.

00:09:43.260 --> 00:09:44.670
We know these two coordinates.

00:09:44.670 --> 00:09:46.360
We know these two coordinates.

00:09:46.360 --> 00:09:49.880
We just need to figure out
this coordinate up here.

00:09:49.880 --> 00:09:54.520
And to do that, let's now
extend into the z-direction.

00:09:54.520 --> 00:09:57.640
So if we were to go
straight up from this point.

00:09:57.640 --> 00:10:00.370
So if we were to go straight
up from that center point,

00:10:00.370 --> 00:10:01.460
we just figured out.

00:10:01.460 --> 00:10:04.230
Let's say we go right.

00:10:04.230 --> 00:10:07.870
I could draw a neater
altitude than that.

00:10:07.870 --> 00:10:11.190
So let's say I were to go
straight up from this point

00:10:11.190 --> 00:10:15.720
right here to the top
of this tetrahedron--

00:10:15.720 --> 00:10:17.815
so let's say
tetrahedron, because I

00:10:17.815 --> 00:10:19.520
was saying it a
little bit weird.

00:10:19.520 --> 00:10:22.005
So now the tetrahedron, you
can imagine, looks like this.

00:10:25.210 --> 00:10:27.650
The drawing is the
hard part here.

00:10:27.650 --> 00:10:30.451
So now this is a face
of the tetrahedron.

00:10:30.451 --> 00:10:32.200
I don't even have to
draw the whole thing.

00:10:32.200 --> 00:10:33.480
I think we can visualize it.

00:10:33.480 --> 00:10:35.770
I'm having trouble
drawing it, making

00:10:35.770 --> 00:10:38.430
my hand go in an
awkward position.

00:10:38.430 --> 00:10:42.010
Let me see if I can just
draw it with dotted lines.

00:10:42.010 --> 00:10:43.030
So that side.

00:10:43.030 --> 00:10:45.965
So what I've just drawn here is
this face of the tetrahedron.

00:10:52.320 --> 00:10:58.000
I've just drawn this magenta
face of the tetrahedron right

00:10:58.000 --> 00:10:59.160
over here.

00:10:59.160 --> 00:11:00.760
And I could draw
the back side too.

00:11:00.760 --> 00:11:03.150
I could draw the
backside just like that.

00:11:03.150 --> 00:11:04.610
I could draw it over there.

00:11:04.610 --> 00:11:06.270
And I think it's
pretty clear to you

00:11:06.270 --> 00:11:08.110
that this point
right here is going

00:11:08.110 --> 00:11:12.280
to lie right on top of this
center point over here.

00:11:12.280 --> 00:11:17.080
So we already know a
few of its coordinates.

00:11:17.080 --> 00:11:18.860
We know its x- and y-coordinate.

00:11:18.860 --> 00:11:20.840
We just need to figure
out its z-coordinate.

00:11:20.840 --> 00:11:22.996
So its x-coordinate
is going to be 0.

00:11:22.996 --> 00:11:24.870
It's going to be the
same x-coordinate here--

00:11:24.870 --> 00:11:27.040
we haven't moved
along the x-axis.

00:11:27.040 --> 00:11:31.711
Its y-coordinate is going to
be the square root of 3 over 3.

00:11:31.711 --> 00:11:33.210
Same as this guy
over here, but it's

00:11:33.210 --> 00:11:35.640
going to have some height
in the z-direction.

00:11:35.640 --> 00:11:37.530
So we need to figure
out what that is.

00:11:37.530 --> 00:11:41.150
And to do that,
what I'm going to do

00:11:41.150 --> 00:11:43.060
is to draw,
essentially, an altitude

00:11:43.060 --> 00:11:44.930
for the face of this front face.

00:11:44.930 --> 00:11:48.780
I'm going to go from this point
up to here, or from this point

00:11:48.780 --> 00:11:50.569
up to here.

00:11:50.569 --> 00:11:52.610
So if you look at this
blue line right over here,

00:11:52.610 --> 00:11:54.800
we already figured out
what this distance is.

00:11:54.800 --> 00:11:56.940
All the faces-- it's a
regular tetrahedron--

00:11:56.940 --> 00:11:59.630
so all of the faces
have the same length.

00:11:59.630 --> 00:12:01.490
So this blue distance
right over here

00:12:01.490 --> 00:12:04.800
is going to be the same as
this distance over here,

00:12:04.800 --> 00:12:07.080
because this was the altitude
for that bottom face.

00:12:07.080 --> 00:12:09.780
And we already figured out
that this distance over here

00:12:09.780 --> 00:12:11.000
is square root of 3.

00:12:11.000 --> 00:12:12.660
We did that, that
was the first thing

00:12:12.660 --> 00:12:14.850
we did that this
distance over here

00:12:14.850 --> 00:12:17.560
from this point to this
point is square root of 3.

00:12:17.560 --> 00:12:20.290
So this distance is also
the square root of 3.

00:12:20.290 --> 00:12:24.150
We also know that this
distance right over here--

00:12:24.150 --> 00:12:26.210
let me do this in a
new color-- we also

00:12:26.210 --> 00:12:28.120
know that this
distance over here

00:12:28.120 --> 00:12:31.460
is the square root of 3 over 3.

00:12:31.460 --> 00:12:32.660
This is a right angle.

00:12:32.660 --> 00:12:34.826
It might not be completely
obvious the way I did it,

00:12:34.826 --> 00:12:35.980
but this was an altitude.

00:12:35.980 --> 00:12:39.920
So now we can figure
out the height

00:12:39.920 --> 00:12:43.130
of the pyramid, or
the tetrahedron.

00:12:43.130 --> 00:12:46.990
And that height to this point
is going to be our z-coordinate.

00:12:46.990 --> 00:12:49.760
So let's call that height h.

00:12:49.760 --> 00:12:51.730
So we know from the
Pythagorean theorem,

00:12:51.730 --> 00:12:57.080
h squared plus the base
squared right over here-- which

00:12:57.080 --> 00:12:59.240
is going to be square
root of 3 over 3 squared

00:12:59.240 --> 00:13:04.410
is going to be 3 over
3 over 9 is going

00:13:04.410 --> 00:13:06.220
to be equal to the
square root of 3

00:13:06.220 --> 00:13:08.810
squared-- it's going
to be equal to 3.

00:13:08.810 --> 00:13:10.900
Or subtracting this
from both sides,

00:13:10.900 --> 00:13:16.450
we have h squared is equal to--
I'll write the-- it's 3 minus 3

00:13:16.450 --> 00:13:21.020
over 9, 3 minus one third,
which is the same thing as 9/3

00:13:21.020 --> 00:13:25.297
minus one third,
which is equal to 8/3.

00:13:25.297 --> 00:13:27.630
And so if we want to take the
square root of both sides,

00:13:27.630 --> 00:13:30.250
we'll get h is equal
to the square root of 8

00:13:30.250 --> 00:13:32.250
is a square root of 4
times square root of 2.

00:13:32.250 --> 00:13:37.260
So that's 2 square roots of
2 over the square root of 3.

00:13:37.260 --> 00:13:39.080
So that is h.

00:13:39.080 --> 00:13:41.500
And that's also the
z-coordinate here.

00:13:41.500 --> 00:13:49.500
So this is 2 square roots of
2 over the square root of 3.

00:13:49.500 --> 00:13:51.759
I'm already close
to 14 minutes here,

00:13:51.759 --> 00:13:53.550
so I'm going to continue
in the next video.

00:13:53.550 --> 00:13:56.470
But we now have the
coordinate for this top point.

00:13:56.470 --> 00:13:58.890
So we can take the average
of these three coordinates,

00:13:58.890 --> 00:14:02.300
find the coordinate for
this point right over here,

00:14:02.300 --> 00:14:04.850
then find the distance between
this point and this point

00:14:04.850 --> 00:14:06.433
that we already got
the coordinate for

00:14:06.433 --> 00:14:08.550
to figure out how
long this side is.

00:14:08.550 --> 00:14:10.270
And then we can prepare
the ratio of that

00:14:10.270 --> 00:14:14.780
to that, cube it to get
the ratio of the volumes.


2003 AIME II problem 4 (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.380
Where we left off
in the last video,

00:00:02.380 --> 00:00:05.050
we were able to figure
out the coordinates of all

00:00:05.050 --> 00:00:09.662
of the vertices for this face
right here of the tetrahedron.

00:00:09.662 --> 00:00:11.870
And so now we can use those
coordinates to figure out

00:00:11.870 --> 00:00:15.620
the coordinates of the center
of that face, which is also

00:00:15.620 --> 00:00:18.570
one of the vertices of
the smaller tetrahedron.

00:00:18.570 --> 00:00:20.300
When we figure out
that coordinate,

00:00:20.300 --> 00:00:21.758
then we can figure
out the distance

00:00:21.758 --> 00:00:23.560
between that point
and that point,

00:00:23.560 --> 00:00:25.950
or between this
point and this point.

00:00:25.950 --> 00:00:27.720
And that will give
us the length of one

00:00:27.720 --> 00:00:29.680
of the sides of the
small tetrahedron.

00:00:29.680 --> 00:00:31.450
And then we can compare
that to the length

00:00:31.450 --> 00:00:35.310
of the large tetrahedron and
get this ratio right over here.

00:00:35.310 --> 00:00:38.400
So let's get the
coordinates of that point

00:00:38.400 --> 00:00:40.560
or this point right over here.

00:00:40.560 --> 00:00:42.596
Let's get the coordinates
of this thing.

00:00:42.596 --> 00:00:44.970
To do that, we just have to
take the average of the x, y,

00:00:44.970 --> 00:00:49.150
and z-coordinates the
vertices of the face.

00:00:49.150 --> 00:00:51.370
Now for x, you
can see, hopefully

00:00:51.370 --> 00:00:55.310
maybe you see that this
is sitting on the yz axis.

00:00:55.310 --> 00:01:00.600
Actually, let me draw the z-axis
here just for visualization.

00:01:00.600 --> 00:01:03.410
So z could go up like
that, and then this

00:01:03.410 --> 00:01:07.580
would be the negative z
direction would be going down

00:01:07.580 --> 00:01:08.400
like that.

00:01:08.400 --> 00:01:11.300
So this is clearly
lying on the zy plane.

00:01:11.300 --> 00:01:15.830
But even if you
didn't see that, you

00:01:15.830 --> 00:01:18.000
could just average
the x points here.

00:01:18.000 --> 00:01:23.480
So we have negative
1 plus 1 plus 0

00:01:23.480 --> 00:01:27.660
over 3, which is
going to be 0 over 3.

00:01:27.660 --> 00:01:30.066
And then you could
average the y points.

00:01:30.066 --> 00:01:39.090
0 plus 0 plus square root of
3 over 3, all of that over 3.

00:01:39.090 --> 00:01:40.710
And then you can
average the z points.

00:01:40.710 --> 00:01:42.620
Let me do this in
a different color.

00:01:42.620 --> 00:01:43.550
Average the z's.

00:01:43.550 --> 00:01:50.810
0 plus 0 plus 2 square roots
of 2 over the square root of 3,

00:01:50.810 --> 00:01:54.540
all of that over 3.

00:01:54.540 --> 00:01:58.900
Now this first coordinate right
over here, clearly just 0.

00:01:58.900 --> 00:02:00.810
Second coordinate
over here is going

00:02:00.810 --> 00:02:04.754
to be the square root of-- this
is the square root of 3 over 3.

00:02:04.754 --> 00:02:05.670
This was hard to read.

00:02:05.670 --> 00:02:08.250
Square root of 3
over 3 divided by 3.

00:02:08.250 --> 00:02:11.720
So it's the square
root of 3 over 9.

00:02:11.720 --> 00:02:14.560
And then we have 2 times
the square root of 2

00:02:14.560 --> 00:02:17.010
over the square root
of 3 divided by 3.

00:02:17.010 --> 00:02:19.740
So it's going to be 2
times the square root of 2

00:02:19.740 --> 00:02:22.750
over 3 times the
square root of 3.

00:02:22.750 --> 00:02:24.690
This right here
are the coordinates

00:02:24.690 --> 00:02:27.530
for the center of this space
face, this coordinate right

00:02:27.530 --> 00:02:28.430
over here.

00:02:28.430 --> 00:02:30.800
Now using that, we can
now find the distance

00:02:30.800 --> 00:02:32.420
between that point
and that point.

00:02:32.420 --> 00:02:34.350
We already know this
point's coordinates.

00:02:34.350 --> 00:02:37.260
We already know the
center of the base.

00:02:37.260 --> 00:02:39.140
We already know that
right over here.

00:02:39.140 --> 00:02:43.170
So let's figure out this
distance or this distance,

00:02:43.170 --> 00:02:47.490
the length of a side of
the smaller tetrahedron.

00:02:47.490 --> 00:02:48.772
So let's write it.

00:02:48.772 --> 00:02:49.730
Let me do it over here.

00:02:49.730 --> 00:02:53.490
The length of the side of a
smaller tetrahedron squared

00:02:53.490 --> 00:02:56.480
is going to be our difference
in x value squared.

00:02:56.480 --> 00:02:58.910
So this has an x value of 0.

00:02:58.910 --> 00:03:00.760
This has an x value of 0.

00:03:00.760 --> 00:03:05.100
So 0 minus 0 squared plus
the difference in y values.

00:03:05.100 --> 00:03:10.770
Square root of 3 over 9
minus square root of 3 over 3

00:03:10.770 --> 00:03:12.340
minus-- Square root
of 3 over 3, just

00:03:12.340 --> 00:03:13.965
so we have a common
denominator, that's

00:03:13.965 --> 00:03:17.906
the same thing as 3
square roots of 3 over 9.

00:03:17.906 --> 00:03:19.530
I just wanted to
rewrite it this way so

00:03:19.530 --> 00:03:23.160
that I have a common
denominator, squared.

00:03:23.160 --> 00:03:26.040
Plus 2 square roots of 2.

00:03:26.040 --> 00:03:30.410
It's getting messy because I'm--
plus 2 square roots of 2 over 3

00:03:30.410 --> 00:03:33.930
square roots of 3 minus
this guy's z-coordinate.

00:03:33.930 --> 00:03:35.630
Well, this guy was
on the xy plane,

00:03:35.630 --> 00:03:39.290
so it's minus 0,
minus 0 squared.

00:03:39.290 --> 00:03:42.307
So this is going to be equal
to-- let me write it here.

00:03:42.307 --> 00:03:43.890
Let me scroll to the
left a little bit

00:03:43.890 --> 00:03:45.380
so I have some space.

00:03:45.380 --> 00:03:49.550
So we have a length of the
small tetrahedron squared

00:03:49.550 --> 00:03:53.300
is going to be equal to-- well,
this is just going to be 0.

00:03:53.300 --> 00:03:55.960
This right here is negative
2 times the square root

00:03:55.960 --> 00:03:58.570
of 3 over 9 squared.

00:03:58.570 --> 00:04:02.060
So negative 2 times the square
root of 3 over 9 squared

00:04:02.060 --> 00:04:05.862
is 4 times 3.

00:04:05.862 --> 00:04:07.570
Negative 2 times square
root of 3 squared

00:04:07.570 --> 00:04:14.590
is going to be 4 times 3
over 9 squared, over 81.

00:04:14.590 --> 00:04:16.570
So we did that part
right over here.

00:04:16.570 --> 00:04:19.510
We had negative 2 times
square root of 3 over 9.

00:04:19.510 --> 00:04:21.240
Square that, you
get this over here.

00:04:21.240 --> 00:04:23.250
So that turns into that.

00:04:23.250 --> 00:04:25.930
And then this term
right over here,

00:04:25.930 --> 00:04:29.490
this is just going to be 2
root 2 over 3 root 3 squared.

00:04:29.490 --> 00:04:33.860
So it's going to be plus-- so
this is going to be 4 times

00:04:33.860 --> 00:04:39.880
2 over 9 times 3.

00:04:39.880 --> 00:04:41.470
Or another way of
saying it, this

00:04:41.470 --> 00:04:42.970
is going to be equal
to-- let's see.

00:04:42.970 --> 00:04:45.420
Well, the numerator and
denominator are divisible by 3.

00:04:45.420 --> 00:04:48.230
So this is the same
thing as 4 over 27.

00:04:48.230 --> 00:04:53.540
So this first term is
4 over 27 plus what

00:04:53.540 --> 00:05:00.280
we have over here, 4 times 2 is
8 over 27, which is equal to,

00:05:00.280 --> 00:05:04.980
which is the same
thing as 12 over 27.

00:05:04.980 --> 00:05:07.380
You divide the numerator
and denominator by 3.

00:05:07.380 --> 00:05:11.440
This is equal to 4 over 9.

00:05:11.440 --> 00:05:18.340
So the length of the small side
squared is equal to 4 over 9.

00:05:18.340 --> 00:05:20.165
You take the principal
root on both sides,

00:05:20.165 --> 00:05:21.040
positive square root.

00:05:21.040 --> 00:05:22.360
We only care about
positive values

00:05:22.360 --> 00:05:23.950
because we're talking
about distances.

00:05:23.950 --> 00:05:25.475
So the length of
one side is going

00:05:25.475 --> 00:05:27.100
to be equal to the
square root of that,

00:05:27.100 --> 00:05:29.900
which is going to be 2 over 3.

00:05:29.900 --> 00:05:32.130
So this distance right
here, this length right here

00:05:32.130 --> 00:05:35.140
is going to be 2/3, or
this length right here

00:05:35.140 --> 00:05:37.280
is going to be 2/3.

00:05:37.280 --> 00:05:40.070
So what's the ratio of
a length of a big side

00:05:40.070 --> 00:05:42.419
to the length of a small side?

00:05:42.419 --> 00:05:44.710
So the length of a big side
we already figured out the.

00:05:44.710 --> 00:05:46.850
Length of a big side is 2.

00:05:46.850 --> 00:05:48.420
This thing is 2.

00:05:48.420 --> 00:05:50.880
What's the length
of a small side?

00:05:50.880 --> 00:05:53.443
Well, based on all the math
we just did, it is 2/3.

00:05:56.790 --> 00:05:58.490
Now before we even
cube it, we can

00:05:58.490 --> 00:06:01.760
divide the numerator and
the denominator by 2.

00:06:01.760 --> 00:06:04.760
Actually, we can divide the
numerator and the denominator

00:06:04.760 --> 00:06:08.520
by 2, so this becomes 1/2 1/3.

00:06:08.520 --> 00:06:12.330
And 1 over 1/3 is
the same thing as 3.

00:06:12.330 --> 00:06:16.540
So 3 cubed is 27.

00:06:16.540 --> 00:06:20.120
Or I could write it
as 27 to 1, since we

00:06:20.120 --> 00:06:22.170
want to write it as a ratio.

00:06:22.170 --> 00:06:25.850
So when you simplify all this,
you really will just get 27.

00:06:25.850 --> 00:06:30.000
But if you want to express
it as a ratio, it is 27 to 1,

00:06:30.000 --> 00:06:32.620
the volume of the
big tetrahedron

00:06:32.620 --> 00:06:34.790
to the volume of the small.

00:06:34.790 --> 00:06:37.640
So that ratio is m over n.

00:06:37.640 --> 00:06:41.220
This is equal to m over
n where m and n are

00:06:41.220 --> 00:06:42.837
relatively prime
positive integers.

00:06:42.837 --> 00:06:44.420
These are definitely
relatively prime.

00:06:44.420 --> 00:06:46.780
Their only common factor is 1.

00:06:46.780 --> 00:06:48.050
Find m plus n?

00:06:48.050 --> 00:06:55.700
So 27 plus 1 is 28,
so our answer is 28.


2003 AIME II problem 3

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.920
Define a "good word" as
a sequence of letters

00:00:03.920 --> 00:00:07.610
that consists only of
the letters A, B, and C.

00:00:07.610 --> 00:00:10.520
Some of these letters may
not appear in the sequence.

00:00:10.520 --> 00:00:14.486
And in which A is never
immediately followed by B,

00:00:14.486 --> 00:00:17.310
B is never immediately
followed by C,

00:00:17.310 --> 00:00:20.250
and C is never
immediately followed

00:00:20.250 --> 00:00:26.570
by A. How many seven letter
"good words" are there?

00:00:26.570 --> 00:00:28.500
So let's just think
about this a little bit.

00:00:28.500 --> 00:00:31.160
So there's letters with
just A's, B's, and C's.

00:00:31.160 --> 00:00:35.140
And then it could be all
A's, all B's, all C's

00:00:35.140 --> 00:00:37.230
because some letters
might not appear.

00:00:37.230 --> 00:00:39.740
And A is never
immediately followed

00:00:39.740 --> 00:00:45.676
by B. So A can only be followed
by another A or another C.

00:00:45.676 --> 00:00:49.940
B is never immediately
followed by C,

00:00:49.940 --> 00:00:54.390
which means that B could only be
followed by an A or another B.

00:00:54.390 --> 00:00:57.540
And C is never
immediately followed by A.

00:00:57.540 --> 00:01:01.650
So C could only be followed
by another C or a B.

00:01:01.650 --> 00:01:04.019
So how many seven letter
"good words" are there?

00:01:04.019 --> 00:01:05.820
So let's just think
about the places.

00:01:05.820 --> 00:01:12.880
We have seven letters, so
1, 2, 3, 4, 5, 6, 7 letters.

00:01:12.880 --> 00:01:15.200
Now, there's no constraints
on this first letter

00:01:15.200 --> 00:01:17.090
since it's not
following anything.

00:01:17.090 --> 00:01:20.710
So it could be an A
B, or a C. So there's

00:01:20.710 --> 00:01:23.529
three possibilities
for this first letter.

00:01:23.529 --> 00:01:25.820
Now, there's three possibilities
for this first letter.

00:01:25.820 --> 00:01:30.020
But no matter what letter this
is, how many possibilities are

00:01:30.020 --> 00:01:32.680
there for this second
letter over here?

00:01:32.680 --> 00:01:34.890
Well, if this was an
A, the second letter

00:01:34.890 --> 00:01:37.650
could only be an A
or a C because it

00:01:37.650 --> 00:01:40.610
can't be followed by
a B. If this was a B,

00:01:40.610 --> 00:01:43.950
the second letter could
only be a B or an A

00:01:43.950 --> 00:01:46.800
because it can't be followed
by a C. If this was a C,

00:01:46.800 --> 00:01:50.290
the second letter could
only be a B, or a C.

00:01:50.290 --> 00:01:52.570
So no matter what letter
this first letter is,

00:01:52.570 --> 00:01:55.727
the second letter can only
have two possibilities.

00:01:55.727 --> 00:01:57.310
There could only be
two possibilities.

00:01:57.310 --> 00:01:58.737
Another way to
think about it is,

00:01:58.737 --> 00:02:00.820
there's one letter, no
matter what letter this is,

00:02:00.820 --> 00:02:02.800
there's one letter that's
being ruled out here.

00:02:02.800 --> 00:02:04.720
So it could only be
two possibilities.

00:02:04.720 --> 00:02:05.540
Well, the same thing is here.

00:02:05.540 --> 00:02:07.123
We're going to stick
some letter here.

00:02:07.123 --> 00:02:09.860
And no matter what letter
there is over here,

00:02:09.860 --> 00:02:12.700
it's going to rule out
one possibility over here.

00:02:12.700 --> 00:02:14.730
So we're going to have
only two possible letters

00:02:14.730 --> 00:02:17.290
that we can put here, no
matter what letter is there.

00:02:17.290 --> 00:02:20.190
And use the same logic, only
two possibilities there,

00:02:20.190 --> 00:02:24.080
only two possibilities there,
only two possibilities there,

00:02:24.080 --> 00:02:25.690
and then only two
possibilities there.

00:02:25.690 --> 00:02:28.420
So how many total
possibilities do we have?

00:02:28.420 --> 00:02:31.360
Well, 3 times 2 times 2
times 2 times 2 times 2.

00:02:31.360 --> 00:02:34.830
This is 1, 2, 3, 4, 5, 6 2's.

00:02:34.830 --> 00:02:38.780
So this is equal to 3
times 2 to the sixth power,

00:02:38.780 --> 00:02:43.630
which is 3 times 2 to
the six is 32 times 2

00:02:43.630 --> 00:02:50.110
is 64, which is equal to
180 plus 12 is equal to 192.

00:02:50.110 --> 00:02:53.810
There's 192 possible
good seven letter

00:02:53.810 --> 00:02:57.960
"good words," where good
words is defined above.


2003 AIME II problem 1

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.700
I got this problem here
from the 2003 AIME exam.

00:00:04.700 --> 00:00:08.360
That stands for the American
Invitational Mathematics Exam.

00:00:08.360 --> 00:00:10.880
And this was actually the
first problem in the exam.

00:00:10.880 --> 00:00:17.230
The product N of three positive
integers is 6 times their sum,

00:00:17.230 --> 00:00:20.060
and one of the integers is
the sum of the other two.

00:00:20.060 --> 00:00:24.080
Find the sum of all
possible values of N.

00:00:24.080 --> 00:00:27.380
So we have to deal with
three positive integers.

00:00:27.380 --> 00:00:30.650
So we have three positive
integers right over here.

00:00:30.650 --> 00:00:33.080
So let's just think about
three positive integers.

00:00:33.080 --> 00:00:35.405
Let's call them a, b, and c.

00:00:35.405 --> 00:00:36.280
They're all positive.

00:00:36.280 --> 00:00:37.530
They're all integers.

00:00:37.530 --> 00:00:41.160
The product N of these
three positive integers--

00:00:41.160 --> 00:00:48.030
so a times b times c is equal
to N-- is 6 times their sum.

00:00:48.030 --> 00:00:51.110
This is equal to
6 times the sum.

00:00:51.110 --> 00:00:52.810
Let me do this in another color.

00:00:52.810 --> 00:00:54.450
So this is their product.

00:00:54.450 --> 00:00:57.690
So the product N of
three positive integers

00:00:57.690 --> 00:01:02.050
is 6 times their sum.

00:01:02.050 --> 00:01:05.310
So this is equal to 6
times the sum of those

00:01:05.310 --> 00:01:09.510
integers a plus b plus c.

00:01:09.510 --> 00:01:13.020
And one of the integers is
the sum of the other two.

00:01:19.950 --> 00:01:23.440
Well, let's just pick c
to be the sum of a and b.

00:01:23.440 --> 00:01:24.190
It doesn't matter.

00:01:24.190 --> 00:01:26.600
These are just names, and
we haven't said one of them

00:01:26.600 --> 00:01:28.330
is larger or less
than the other one.

00:01:28.330 --> 00:01:31.400
So let's just say
that a plus b is

00:01:31.400 --> 00:01:33.880
equal to c, that
one of the integers

00:01:33.880 --> 00:01:36.990
is the sum of the other two.
c is the sum of a plus b.

00:01:36.990 --> 00:01:41.990
Find the sum of all
possible values of N.

00:01:41.990 --> 00:01:45.160
So let's just try to do a
little bit of manipulation

00:01:45.160 --> 00:01:47.050
of the information we
have here, and maybe we

00:01:47.050 --> 00:01:50.146
can get some relationship
or some constraints

00:01:50.146 --> 00:01:51.770
on our numbers, and
then we can kind of

00:01:51.770 --> 00:01:54.050
go through all of
the possibilities.

00:01:54.050 --> 00:01:56.730
So let's see, we know that
a plus b is equal to c.

00:01:56.730 --> 00:02:02.360
So we can replace c
everywhere with a plus b.

00:02:02.360 --> 00:02:04.110
So this expression
right over here

00:02:04.110 --> 00:02:09.130
becomes ab, which is
just a times b, times c.

00:02:09.130 --> 00:02:11.780
But instead of c, I'm going to
write an a plus b over here.

00:02:15.500 --> 00:02:25.440
And then that is equal to
6 times a plus b plus c.

00:02:25.440 --> 00:02:31.360
And so once again, I'll replace
with the c with an a plus b,

00:02:31.360 --> 00:02:33.610
and then what does
this simplify to?

00:02:33.610 --> 00:02:36.150
So on the right-hand side,
we have 6 times a plus b

00:02:36.150 --> 00:02:37.020
plus a plus b.

00:02:37.020 --> 00:02:43.680
This is the same thing
as 6 times 2a plus 2b,

00:02:43.680 --> 00:02:45.520
just added the a's and the b's.

00:02:45.520 --> 00:02:46.700
And we can factor out a 2.

00:02:46.700 --> 00:02:49.740
This is the same thing as if
you take out a 2, 6 times 2

00:02:49.740 --> 00:02:53.320
is 12 times a plus b.

00:02:53.320 --> 00:02:55.856
The left-hand side
right over here

00:02:55.856 --> 00:03:01.720
is still a times b
or ab times a plus b.

00:03:01.720 --> 00:03:07.710
So ab times a plus b has got to
be equal to 12 times a plus b.

00:03:07.710 --> 00:03:09.430
So this is pretty
interesting here.

00:03:09.430 --> 00:03:12.610
We can divide both
sides by a plus b.

00:03:12.610 --> 00:03:17.140
We know that a plus b cannot be
equal to 0 since all of these

00:03:17.140 --> 00:03:20.940
numbers have to be
positive numbers.

00:03:20.940 --> 00:03:24.740
And the reason why I say that
is if it was 0, dividing by 0

00:03:24.740 --> 00:03:27.450
would give you an
undefined answer.

00:03:27.450 --> 00:03:30.130
So if we divide both
sides by a plus b,

00:03:30.130 --> 00:03:34.150
we get a times b is equal to 12.

00:03:34.150 --> 00:03:36.140
So all the constraints
that they gave us

00:03:36.140 --> 00:03:38.290
boiled down to this
right over here.

00:03:38.290 --> 00:03:41.530
The product of a and
b is equal to 12.

00:03:41.530 --> 00:03:43.726
And there's only
so many numbers, so

00:03:43.726 --> 00:03:46.100
many positive integers where
you if you take the product,

00:03:46.100 --> 00:03:46.950
you get 12.

00:03:46.950 --> 00:03:49.174
Let's try them out.

00:03:49.174 --> 00:03:50.590
So let me write
some columns here.

00:03:50.590 --> 00:03:54.290
Let's say a, b, c.

00:03:54.290 --> 00:04:00.070
And then we care
about their product,

00:04:00.070 --> 00:04:03.700
so I'll write that
over here, so abc.

00:04:03.700 --> 00:04:08.050
So if a is 1, b
is going to be 12.

00:04:08.050 --> 00:04:13.530
c is the sum of those two, so
c is going to be 13, 1 times

00:04:13.530 --> 00:04:15.380
12 times 13.

00:04:15.380 --> 00:04:21.959
12 times 12 is 144 plus
another 12 is going to be 156.

00:04:21.959 --> 00:04:24.620
And just for fun, you
can verify that this

00:04:24.620 --> 00:04:27.035
is going to be equal
to 6 times their sum.

00:04:27.035 --> 00:04:32.280
Their sum is 26,
26 times 6 is 156.

00:04:32.280 --> 00:04:33.530
So this one definitely worked.

00:04:33.530 --> 00:04:34.850
It definitely worked
for the constraints.

00:04:34.850 --> 00:04:37.100
And it should because we
boiled down those constraints

00:04:37.100 --> 00:04:40.060
to a times b need
to be equal to 12.

00:04:40.060 --> 00:04:41.720
So let's try another one.

00:04:41.720 --> 00:04:45.670
2 times 6, their sum is 8.

00:04:45.670 --> 00:04:48.200
And then if I were to take
the product of all of these,

00:04:48.200 --> 00:04:52.730
you get 2 times 6
is 12 times 8 is 96.

00:04:55.370 --> 00:04:58.830
And then we could try 3 and 4.

00:04:58.830 --> 00:05:01.154
3 plus 4 is 7.

00:05:01.154 --> 00:05:06.705
3 times 4 is 12 times 7.

00:05:06.705 --> 00:05:09.080
Actually, I should have known
the a times b is always 12,

00:05:09.080 --> 00:05:11.560
so you just have to multiply
12 times this last column.

00:05:11.560 --> 00:05:14.190
12 times 7 is 84.

00:05:17.080 --> 00:05:19.199
And there aren't any others.

00:05:19.199 --> 00:05:21.240
You definitely can't go
above 12 because then you

00:05:21.240 --> 00:05:22.580
would have to deal
with non-integers.

00:05:22.580 --> 00:05:23.840
You would have to
deal with fractions.

00:05:23.840 --> 00:05:25.520
You can't do the negative
versions of these

00:05:25.520 --> 00:05:27.436
because they all have
to be positive integers.

00:05:27.436 --> 00:05:28.120
So that's it.

00:05:28.120 --> 00:05:30.620
Those are all of the
possible positive integers

00:05:30.620 --> 00:05:33.070
where if you take their
products you get 12.

00:05:33.070 --> 00:05:35.300
We've essentially
just factored 12.

00:05:35.300 --> 00:05:41.620
So they want us to find the sum
of all possible values of N.

00:05:41.620 --> 00:05:43.420
Well, these are all
the possible values

00:05:43.420 --> 00:05:45.350
of N. N was the product
of those integers.

00:05:45.350 --> 00:05:47.920
So let's just take the sum.

00:05:47.920 --> 00:05:52.980
6 plus 6 is 12 plus
4 is 16, 1 plus 5

00:05:52.980 --> 00:06:01.740
is 6 plus 9 is 15 plus
8 is 23, 2 plus 1 is 3.

00:06:01.740 --> 00:06:04.750
So our answer is 336.


Simple Differential Equation Example

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.290
Let f be a real valued
differentiable function

00:00:04.290 --> 00:00:08.150
on R-- the set of all real
numbers-- such that f of 1

00:00:08.150 --> 00:00:09.850
is equal to 1.

00:00:09.850 --> 00:00:14.680
If the y-intercept of the
tangent at any point P-- which

00:00:14.680 --> 00:00:16.900
has the coordinates x
and y-- on the curve y

00:00:16.900 --> 00:00:19.990
equals f of x is equal to
the cube of the abscissa--

00:00:19.990 --> 00:00:23.160
I haven't seen the word
abscissa in a while-- of P,

00:00:23.160 --> 00:00:26.420
then the value of f of
negative 3 is equal to.

00:00:26.420 --> 00:00:28.170
So let's just decode
what they're saying.

00:00:28.170 --> 00:00:31.120
So let me draw a
potential f right here.

00:00:31.120 --> 00:00:33.110
So that's our y-axis.

00:00:33.110 --> 00:00:37.080
And then, let's call
that our x-axis.

00:00:37.080 --> 00:00:39.830
And let me just draw a
simple, potential f of x.

00:00:39.830 --> 00:00:42.070
So let's say it looks
something like that.

00:00:42.070 --> 00:00:42.900
I don't know what it looks like.

00:00:42.900 --> 00:00:44.274
We don't know what
it looks like.

00:00:44.274 --> 00:00:47.040
We just know it's some real
valued differentiable function

00:00:47.040 --> 00:00:49.730
on R. So that is f of x.

00:00:49.730 --> 00:00:52.610
So what is the tangent
going to look like?

00:00:52.610 --> 00:01:00.360
So let's say we have some point
here, which is x, 0 and y, 0.

00:01:00.360 --> 00:01:01.720
So it's a particular x and y.

00:01:01.720 --> 00:01:03.810
What would the tangent
there look like?

00:01:03.810 --> 00:01:05.300
Well, it would be
a line that has

00:01:05.300 --> 00:01:08.440
the same slope as the
function does at that point.

00:01:08.440 --> 00:01:11.710
So it would be a line.

00:01:11.710 --> 00:01:13.890
I can draw a better
version than that.

00:01:13.890 --> 00:01:15.960
It would be a line
that looks like that.

00:01:19.560 --> 00:01:22.290
This line right here
would have the equation

00:01:22.290 --> 00:01:27.760
y is equal to mx plus
b, where m is the slope

00:01:27.760 --> 00:01:30.220
and b is the y-intercept.

00:01:30.220 --> 00:01:31.970
Now, we know that
the slope is going

00:01:31.970 --> 00:01:35.160
to be the same as the slope
at this point in the function.

00:01:35.160 --> 00:01:37.990
And what is the slope at
that point in the function?

00:01:37.990 --> 00:01:40.010
Well, the slope at that
point in that function

00:01:40.010 --> 00:01:42.440
is the derivative
of the function.

00:01:42.440 --> 00:01:48.040
So this is going to
be f prime of x, 0.

00:01:48.040 --> 00:01:50.340
So we take the derivative
of valuated x, 0.

00:01:50.340 --> 00:01:54.510
That will give us the
slope right at that point.

00:01:54.510 --> 00:01:57.620
So this is going to
be y is equal to f

00:01:57.620 --> 00:02:02.500
prime of x, 0 times x plus b.

00:02:02.500 --> 00:02:06.780
Now, if we wanted to solve
for the y-intercept-- for b

00:02:06.780 --> 00:02:09.120
right there-- we
just have to find out

00:02:09.120 --> 00:02:10.539
some point that's on this line.

00:02:10.539 --> 00:02:12.770
And we know a point that's
on the line. x, 0 and y,

00:02:12.770 --> 00:02:14.540
0 is on the line.

00:02:14.540 --> 00:02:17.280
So let's substitute x, 0
and y, 0 into this equation

00:02:17.280 --> 00:02:18.990
and then solve for b.

00:02:18.990 --> 00:02:29.960
So we have y, 0 is
equal to f prime

00:02:29.960 --> 00:02:32.910
of x, 0-- that's just the
slope-- times x, 0-- we're just

00:02:32.910 --> 00:02:40.220
substituting y, 0 and x, 0
into this equation-- plus b.

00:02:40.220 --> 00:02:43.350
So I just put x, 0 and y, 0 into
this equation to solve for b.

00:02:43.350 --> 00:02:45.460
If you solve for b, you
could just subtract this

00:02:45.460 --> 00:02:47.330
from both sides of the equation.

00:02:47.330 --> 00:02:49.055
We get b.

00:02:49.055 --> 00:02:59.690
b is equal to y, 0 minus f
prime of x, 0 times x, 0.

00:02:59.690 --> 00:03:02.760
Now, that's the
y-intercept of the tangent.

00:03:02.760 --> 00:03:05.770
They tell us the y-intercept of
the tangent at any point P x,

00:03:05.770 --> 00:03:08.060
y on the curve y
equals f of x is

00:03:08.060 --> 00:03:11.940
equal to the cube of
the abscissa of P.

00:03:11.940 --> 00:03:16.360
An abscissa is just a fancy
way of saying the x-coordinate.

00:03:16.360 --> 00:03:17.860
Clearly, they're
just using the word

00:03:17.860 --> 00:03:20.320
to filter out people who don't
know the word "abscissa."

00:03:20.320 --> 00:03:21.940
Abscissa just means
the x-coordinate.

00:03:21.940 --> 00:03:26.230
If I tell you I have the point
2 comma 3, the abscissa of this

00:03:26.230 --> 00:03:27.320
is just 2.

00:03:27.320 --> 00:03:29.220
It's just the x-coordinate.

00:03:29.220 --> 00:03:32.670
So all this sentence is telling
us is that the y-intercept is

00:03:32.670 --> 00:03:35.110
equal to the cube
of the x-coordinate.

00:03:35.110 --> 00:03:37.020
So the y-intercept--
it's this-- is

00:03:37.020 --> 00:03:39.670
going to be equal to the
cube of the x-coordinate.

00:03:39.670 --> 00:03:42.090
The x-coordinate here is x, 0.

00:03:42.090 --> 00:03:45.070
So it's going to be
equal to x, 0 cubed.

00:03:45.070 --> 00:03:51.690
So for any x, 0 and y,
0 for any x, 0 and y,

00:03:51.690 --> 00:03:56.010
0 that's on the function itself,
we now get this expression.

00:03:56.010 --> 00:03:57.480
I'll do it in a new color.

00:03:57.480 --> 00:04:03.060
We get y, 0 minus f
prime of x, 0 times x,

00:04:03.060 --> 00:04:05.640
0 is equal to x, 0 to the third.

00:04:05.640 --> 00:04:07.537
Now, these are
particular x's and y's.

00:04:07.537 --> 00:04:08.870
It's just for a case of example.

00:04:08.870 --> 00:04:10.070
It applies to all of them.

00:04:10.070 --> 00:04:13.950
So we could say it applies to
all of the points on the curve.

00:04:13.950 --> 00:04:16.209
So we could say, for
any point on the curve,

00:04:16.209 --> 00:04:22.170
we get y minus f
prime of x times x

00:04:22.170 --> 00:04:25.985
is equal to x to the third.

00:04:25.985 --> 00:04:29.530
And we know that y
is equal to f of x.

00:04:29.530 --> 00:04:31.110
Or another way of
thinking about it

00:04:31.110 --> 00:04:34.760
is that y prime is
equal to f prime of x.

00:04:34.760 --> 00:04:35.719
So we could write this.

00:04:35.719 --> 00:04:37.510
And what we're getting
here is a little bit

00:04:37.510 --> 00:04:38.670
of a differential equation.

00:04:38.670 --> 00:04:46.200
We get y minus y prime times
x is equal to x to the third.

00:04:46.200 --> 00:04:46.700
Let's see.

00:04:46.700 --> 00:04:50.270
We can could divide both
sides of this equation by x.

00:04:50.270 --> 00:04:55.300
We get y over x minus y
prime is equal to x squared--

00:04:55.300 --> 00:04:57.580
just divided everything by x.

00:04:57.580 --> 00:05:00.430
And then we can subtract
this from both sides

00:05:00.430 --> 00:05:02.380
and then add this
to both sides--

00:05:02.380 --> 00:05:04.140
kind of skipping a
step right there.

00:05:04.140 --> 00:05:07.390
So we get y over x-- actually,
let me write it this way.

00:05:07.390 --> 00:05:09.500
I don't want to skip steps.

00:05:09.500 --> 00:05:12.140
So let's subtract y
over x from both sides.

00:05:12.140 --> 00:05:16.480
We get negative y prime is
equal to negative y over x

00:05:16.480 --> 00:05:17.942
plus x squared.

00:05:17.942 --> 00:05:19.900
And then, multiply both
sides times negative 1.

00:05:19.900 --> 00:05:26.050
We get y prime is equal to
y over x minus x squared.

00:05:26.050 --> 00:05:30.160
And so a little bit of a
differential equation here.

00:05:30.160 --> 00:05:32.090
And solving any
differential equation

00:05:32.090 --> 00:05:33.474
is really a bit of an art.

00:05:33.474 --> 00:05:34.890
And so we have to
think about it--

00:05:34.890 --> 00:05:37.380
if we could separate the y's
from the x's-- if we could put

00:05:37.380 --> 00:05:40.380
the y's all on this side--
then it would be pretty easy

00:05:40.380 --> 00:05:42.724
to solve, but it's
not so obvious.

00:05:42.724 --> 00:05:44.140
And just to give
an example-- just

00:05:44.140 --> 00:05:46.530
to inform our intuition--
if this was a simpler

00:05:46.530 --> 00:05:50.670
version, if this was
just dy/dx-- which

00:05:50.670 --> 00:05:52.870
is the same thing as y
prime-- if it was just

00:05:52.870 --> 00:05:55.270
equal to y over x,
if it was just that,

00:05:55.270 --> 00:05:57.890
so if this negative x
squared term wasn't there,

00:05:57.890 --> 00:06:00.550
then this would be a separable
differential equation.

00:06:00.550 --> 00:06:04.610
You could divide both sides by
y and multiply both sides times

00:06:04.610 --> 00:06:05.590
dx.

00:06:05.590 --> 00:06:08.740
So if you multiplied
this times dx over y

00:06:08.740 --> 00:06:13.040
and you multiplied this times dx
over y, these would cancel out.

00:06:13.040 --> 00:06:14.000
Those would cancel out.

00:06:14.000 --> 00:06:18.070
And you would get ydy
is equal to x-- sorry.

00:06:18.070 --> 00:06:18.940
Not ydy.

00:06:18.940 --> 00:06:20.752
You'd get 1 over y.

00:06:20.752 --> 00:06:25.630
1 over y dy is equal
to 1 over x dx.

00:06:25.630 --> 00:06:27.510
You integrate both
sides of that.

00:06:35.010 --> 00:06:36.930
You would get the
natural log of y--

00:06:36.930 --> 00:06:39.130
that's the antiderivative
of 1 over y--

00:06:39.130 --> 00:06:46.830
is equal to the natural log
of x plus some constant.

00:06:46.830 --> 00:06:50.400
And we could write any constant
as plus the natural log

00:06:50.400 --> 00:06:51.640
of some constant.

00:06:51.640 --> 00:06:54.840
It could just be any
constant over there.

00:06:54.840 --> 00:06:58.370
And then this is the same
thing as the natural log of y

00:06:58.370 --> 00:07:02.860
is equal to the natural log
of-- when you add exponents

00:07:02.860 --> 00:07:05.270
like this, that's essentially
what a logarithm is.

00:07:05.270 --> 00:07:10.760
It's essentially taking a
product of these two terms.

00:07:10.760 --> 00:07:15.150
This is equal to cx or
that y is equal to cx.

00:07:15.150 --> 00:07:17.730
Now, this is all just to
inform our intuition here.

00:07:17.730 --> 00:07:20.160
If this negative x
squared term wasn't here,

00:07:20.160 --> 00:07:22.070
then we would be able
solve it pretty easily.

00:07:22.070 --> 00:07:24.650
y would be equal to cx.

00:07:24.650 --> 00:07:27.490
If this term wasn't here, taking
the antiderivative of this

00:07:27.490 --> 00:07:28.730
is pretty straightforward.

00:07:28.730 --> 00:07:32.335
It would be negative 1/3
x to the third power.

00:07:35.180 --> 00:07:38.280
Our intuition here is--
maybe some combination

00:07:38.280 --> 00:07:40.180
of the antiderivatives
of this and this--

00:07:40.180 --> 00:07:41.805
and you can't just
take them separately

00:07:41.805 --> 00:07:43.380
because this y is
going to have to be

00:07:43.380 --> 00:07:45.400
equal to the antiderivative
of the whole thing.

00:07:45.400 --> 00:07:48.080
It's not just the antiderivative
of this part over here.

00:07:48.080 --> 00:07:49.740
But if this guy
wasn't here, then it

00:07:49.740 --> 00:07:50.979
could be y is equal to cx.

00:07:50.979 --> 00:07:53.020
If this guy wasn't here,
you would have something

00:07:53.020 --> 00:07:54.450
to the third power.

00:07:54.450 --> 00:07:57.080
So just to try things out-- and
solving differential equations

00:07:57.080 --> 00:08:00.220
is always a bit
of an art-- let's

00:08:00.220 --> 00:08:01.999
just say that y is
equal to-- we know

00:08:01.999 --> 00:08:03.540
we're going to have
to have something

00:08:03.540 --> 00:08:04.720
to the third power in there.

00:08:04.720 --> 00:08:07.610
So let's call it Ax
to the third power.

00:08:07.610 --> 00:08:12.650
And we know we're going to
have a x power in there.

00:08:12.650 --> 00:08:14.690
Let's call it plus Bx.

00:08:14.690 --> 00:08:16.900
And then, let's call it plus C.

00:08:16.900 --> 00:08:19.190
And that's our intuition--
that somehow that this

00:08:19.190 --> 00:08:21.687
will help you get to this
term right over here.

00:08:21.687 --> 00:08:23.770
And then, this term-- when
I take the derivative--

00:08:23.770 --> 00:08:26.190
will help to get
that term over there.

00:08:26.190 --> 00:08:28.310
So what would be the
derivative of this?

00:08:28.310 --> 00:08:30.900
The derivative of this--
y prime, in this case--

00:08:30.900 --> 00:08:34.860
would be 3Ax squared plus B.

00:08:34.860 --> 00:08:39.409
And now, what would y over x
be? y over x would be equal to--

00:08:39.409 --> 00:08:41.419
notice, I'm just
saying y prime is this.

00:08:41.419 --> 00:08:50.480
y over x would be Ax squared
plus B plus C over x.

00:08:50.480 --> 00:08:53.665
This differential equation
here would reduce to--

00:08:53.665 --> 00:08:55.490
and we just suspect
that this might

00:08:55.490 --> 00:08:59.620
be a solution-- this y prime
is the same thing as this.

00:08:59.620 --> 00:09:05.930
So we have 3Ax squared plus
B is equal to y over x.

00:09:05.930 --> 00:09:11.110
It's equal to this thing
over here-- Ax squared plus B

00:09:11.110 --> 00:09:15.063
plus C over x minus
this x squared.

00:09:20.140 --> 00:09:22.180
So what does this simplify to?

00:09:22.180 --> 00:09:24.240
If we wanted to
solve this equation,

00:09:24.240 --> 00:09:26.034
we could combine
the x squared term.

00:09:26.034 --> 00:09:27.200
So let me write it this way.

00:09:27.200 --> 00:09:32.540
We have 3Ax squared
plus B is equal

00:09:32.540 --> 00:09:35.400
to-- we have Ax squared
minus x squared.

00:09:35.400 --> 00:09:41.960
So that's equal to A minus 1.

00:09:41.960 --> 00:09:45.385
This is equal to A
minus 1 x squared.

00:09:49.060 --> 00:09:53.730
And then, I have
plus B plus C over x.

00:09:53.730 --> 00:09:56.780
This is what the right-hand
side simplifies to,

00:09:56.780 --> 00:09:59.300
and this is what the
left-hand side is.

00:09:59.300 --> 00:10:03.680
In this, we have no constrain
on B. B so far can be anything.

00:10:03.680 --> 00:10:04.995
B could be anything.

00:10:07.990 --> 00:10:10.350
There's no C term over here.

00:10:10.350 --> 00:10:12.210
And actually, more
importantly, there's no 1

00:10:12.210 --> 00:10:13.400
over x term over here.

00:10:13.400 --> 00:10:15.610
So C would have
to be equal to 0.

00:10:15.610 --> 00:10:18.940
There's no C. C is equal to 0.

00:10:18.940 --> 00:10:20.880
This term does not exist.

00:10:20.880 --> 00:10:22.570
There is no C.

00:10:22.570 --> 00:10:24.500
And then, when you
look at this, this

00:10:24.500 --> 00:10:27.917
coefficient-- the 3A-- the
coefficient of the x squared

00:10:27.917 --> 00:10:30.250
on the left-hand side has to
be equal to the coefficient

00:10:30.250 --> 00:10:33.330
of the x squared on the
right-hand side in order

00:10:33.330 --> 00:10:34.590
for these 2 to be equal.

00:10:34.590 --> 00:10:38.370
So we'd get 3A is
equal to A minus 1.

00:10:38.370 --> 00:10:40.020
Subtract A from both sides.

00:10:40.020 --> 00:10:42.200
2A is equal to negative 1.

00:10:42.200 --> 00:10:44.670
A is equal to negative 1/2.

00:10:44.670 --> 00:10:45.840
So our hunch paid off.

00:10:45.840 --> 00:10:51.080
We were able to figure
out A's, B's, and C's

00:10:51.080 --> 00:10:56.620
so that this function does
satisfy our little differential

00:10:56.620 --> 00:10:57.420
equation.

00:10:57.420 --> 00:11:01.190
And our value we got
is y is equal to-- I

00:11:01.190 --> 00:11:02.450
keep using that green color.

00:11:02.450 --> 00:11:03.510
I don't want to.

00:11:03.510 --> 00:11:10.300
We get y is equal to
negative 1/2 x to the third.

00:11:10.300 --> 00:11:14.250
That's our value for A.
Plus, B could be anything.

00:11:14.250 --> 00:11:17.870
Plus Bx.

00:11:17.870 --> 00:11:20.160
So this is our value for f of x.

00:11:20.160 --> 00:11:22.850
And they gave us
some information.

00:11:22.850 --> 00:11:26.790
They told us in the problem
that f of 1 is equal to 1.

00:11:26.790 --> 00:11:30.390
We can use that information
to solve for B. f of 1

00:11:30.390 --> 00:11:32.240
is equal to 1.

00:11:32.240 --> 00:11:35.930
So when x is equal to
1, y is equal to 1.

00:11:35.930 --> 00:11:40.160
So we get 1 is equal
to negative 1/2 times 1

00:11:40.160 --> 00:11:46.980
plus B times 1 is just B. So
we can add 1/2 to both sides.

00:11:46.980 --> 00:11:50.240
And we get 3/2 is equal to B.

00:11:50.240 --> 00:11:54.680
So we're now able to fully
describe our function.

00:11:54.680 --> 00:12:02.920
Our function is y is
equal to negative 1/2

00:12:02.920 --> 00:12:07.995
x to the third plus 3/2x.

00:12:11.760 --> 00:12:14.100
So that's our function,
but that's not

00:12:14.100 --> 00:12:15.096
what they're asking us.

00:12:15.096 --> 00:12:16.720
They're not asking
us for the function.

00:12:16.720 --> 00:12:19.165
They're asking us for the
value of f of negative 3.

00:12:22.100 --> 00:12:26.060
f of negative 3 is equal
to negative 1/2 times

00:12:26.060 --> 00:12:31.370
negative 3 to the third is
negative 27 plus 3/2 times

00:12:31.370 --> 00:12:32.090
negative 3.

00:12:32.090 --> 00:12:37.040
So minus 9/2.

00:12:37.040 --> 00:12:41.170
This will simplify to--
well, everything's over 2.

00:12:41.170 --> 00:12:43.580
Negative 27 is positive 27.

00:12:43.580 --> 00:12:45.320
And then you have a minus 9.

00:12:45.320 --> 00:12:48.270
This is 27 over 2, 9 over 2.

00:12:48.270 --> 00:12:49.160
negative 9 over 2.

00:12:49.160 --> 00:12:54.650
So this is equal to 18 over
2, which is equal to 9.

00:12:54.650 --> 00:12:56.180
And we are done.

00:12:56.180 --> 00:12:59.090
This is equal to 9.


Trigonometric system example

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.030
The number of all possible values of theta,

00:00:03.030 --> 00:00:04.890
where theta is between zero and pi

00:00:05.140 --> 00:00:08.140
for which the system of equations

00:00:08.140 --> 00:00:11.380
y plus z cosine of 3 theta is equal x y z sin 3 theta

00:00:11.820 --> 00:00:15.590
x sin of 3 theta is equal to 2 cos 3 theta over y

00:00:15.590 --> 00:00:17.420
plus 2 sin 3 theta over z

00:00:17.420 --> 00:00:21.640
xyz sin 3 theta is equal y plus two z cos of 3 theta

00:00:21.640 --> 00:00:24.500
plus y sin 3 theta, has a solution

00:00:25.080 --> 00:00:27.960
x naught y naught z naught with y naught and z naught

00:00:27.960 --> 00:00:31.920
not equal to zero is, which means either of these cannot equal zero

00:00:32.200 --> 00:00:34.530
So this is a pretty daunting thing,

00:00:34.530 --> 00:00:36.680
I just like to simplify this as much as possible.

00:00:36.680 --> 00:00:41.410
I want to do two things. All of the trigonometric functions here

00:00:41.410 --> 00:00:43.640
they're all - they're taking sin of,

00:00:43.640 --> 00:00:47.370
the cosine of three theta and have theta between zero and pi

00:00:47.370 --> 00:00:48.300
I'll do a little substitution.

00:00:48.300 --> 00:00:52.690
this is just from my brain,cause it simplifies my logic a little bit

00:00:52.690 --> 00:00:54.420
let's make a substitution

00:00:54.420 --> 00:00:57.940
that u is equal to -

00:00:57.940 --> 00:01:00.900
is equal to three theta

00:01:00.900 --> 00:01:06.470
and then the u - when theta is zero, u is going to be -

00:01:06.470 --> 00:01:08.710
so u is still going to be greater than zero

00:01:08.710 --> 00:01:13.660
and when theta is pi, u is going to - have to be less than three pi

00:01:13.660 --> 00:01:19.600
And so - let's find the number of Us between zero and three pi

00:01:19.600 --> 00:01:23.830
where y plus z cosine of u is equal to xyz sine of u

00:01:23.830 --> 00:01:25.950
so on and so forth. so that -

00:01:25.950 --> 00:01:28.440
Let's do that substitution and I'm going to rearrange these

00:01:28.440 --> 00:01:30.370
so that they start at least look a little familiar

00:01:30.370 --> 00:01:35.170
and see if we can somehow start manipulating with the these equations

00:01:35.170 --> 00:01:37.430
and actually try to cancel out terms

00:01:38.260 --> 00:01:40.130
so this first one up here

00:01:40.130 --> 00:01:42.180
just to get in a term ways i can recognize.

00:01:42.180 --> 00:01:44.200
let me distribute the cosine of three theta

00:01:44.200 --> 00:01:47.460
or which will now call the cosine of u. So let me just writ this

00:01:47.460 --> 00:01:48.680
This is u

00:01:48.680 --> 00:01:57.130
this is u, this u now, this is u, u, u, u and u

00:01:57.130 --> 00:01:59.100
so this - if we distribute the cosine of u,

00:01:59.100 --> 00:02:05.000
this becomes cosine of u - let me put the y out front.

00:02:05.000 --> 00:02:18.690
so it's y cosine of u plus z cosine of u is equal to xyz sine of u.

00:02:18.690 --> 00:02:20.110
that's this first equation

00:02:20.110 --> 00:02:22.820
this second equation

00:02:22.820 --> 00:02:25.430
this second equation it looks like -

00:02:25.440 --> 00:02:29.320
well if we multiply both sides of this equation by yz

00:02:29.320 --> 00:02:32.650
we're gonna have xyz sine of u on the right hand side

00:02:32.650 --> 00:02:34.260
which is the exact same thing we have here.

00:02:34.260 --> 00:02:41.480
so let's multiply both sides of this equation by xyz, both sides.

00:02:41.480 --> 00:02:46.030
the right hand of the equation - no, not by xyz, just by yz

00:02:46.030 --> 00:02:49.770
So we are gonna multiply both sides of this equation by yz.

00:02:49.770 --> 00:02:52.070
so that these leave the denominator

00:02:52.070 --> 00:02:56.430
we are also gonna multiply the right side by yz

00:02:56.430 --> 00:02:59.210
the left hand side becomes xyz sine of u

00:02:59.210 --> 00:03:01.000
and let me write it over here just write it like this.

00:03:01.000 --> 00:03:05.060
So this is x yz sine of u

00:03:05.060 --> 00:03:10.030
and then yz times two cosine of u is going to be

00:03:10.030 --> 00:03:12.260
two z cosine of u

00:03:12.260 --> 00:03:17.060
so two z the cosine of u, I just swap the sides

00:03:17.060 --> 00:03:20.300
and then yz times two sine of u

00:03:20.300 --> 00:03:23.510
is going to be two y sine of u

00:03:23.510 --> 00:03:28.810
so two y - two y sine of u.

00:03:28.810 --> 00:03:29.910
So this is the second equation.

00:03:29.910 --> 00:03:31.820
now they don't look bad different

00:03:31.820 --> 00:03:34.290
when they were written like this it look very different.

00:03:34.290 --> 00:03:35.060
I'll thinks about this one.

00:03:35.060 --> 00:03:37.260
this side has xyz sine of u

00:03:37.260 --> 00:03:38.910
I'll do it in magenta

00:03:38.910 --> 00:03:41.540
this equation xyz sine of u.I'll write it on this side

00:03:41.540 --> 00:03:49.210
xyz sine of u is equal to -

00:03:49.210 --> 00:03:51.440
is equal to let's see we have a two z

00:03:51.440 --> 00:03:55.760
Let's distribute this cosine of u. we have a two z cosine of u

00:03:55.760 --> 00:03:59.970
so plus two z cosine -

00:03:59.970 --> 00:04:04.200
two z cosine of u plus y cosine of u

00:04:04.200 --> 00:04:09.900
so we have a plus y cosine of u.

00:04:09.900 --> 00:04:13.000
that's that times that plus y sine of u

00:04:13.000 --> 00:04:19.450
so let me scroll to the left a little bit. y sine of u

00:04:19.450 --> 00:04:21.450
so I have rewritten these three equations.

00:04:21.460 --> 00:04:24.160
The problem looks a lot less daunting right now.

00:04:24.160 --> 00:04:31.160
Let's try to figure out the number of Us between zero and three pi

00:04:31.160 --> 00:04:34.590
that will satisfy - that will give us a solution here.

00:04:34.590 --> 00:04:38.120
so let's see all of these three equations are equal to

00:04:38.120 --> 00:04:41.720
this expression right over here. so the left-hand sides -

00:04:41.720 --> 00:04:45.400
the left-hand sides of these equations all have to equal each other

00:04:45.400 --> 00:04:48.690
because they're all equaled the exact same value

00:04:48.690 --> 00:04:53.680
so let's do that. Let's see what we can do in the way of canceling

00:04:53.680 --> 00:04:55.810
in the way of canceling things out.

00:04:55.810 --> 00:04:59.490
Let's see if we use this is a plus right here.

00:04:59.490 --> 00:05:01.490
I don't know I wrote an equal here

00:05:01.490 --> 00:05:04.870
so let's see if we write this has got to be equal to that.

00:05:04.870 --> 00:05:06.870
So let me use these two first

00:05:07.430 --> 00:05:09.000
so this thing

00:05:10.180 --> 00:05:21.530
so we have two y sine of u plus two z cosine of u

00:05:21.530 --> 00:05:24.720
is equal to this which is that, which has to be equal to this,

00:05:24.730 --> 00:05:32.860
is going to be equal to y sine of u plus y cosine of u

00:05:32.860 --> 00:05:39.080
plus two z cosine - plus two z cosine of u

00:05:39.080 --> 00:05:41.220
we have two z cosine of u on both sides,

00:05:41.220 --> 00:05:44.620
so that gets rid of the z terms

00:05:44.620 --> 00:05:47.520
and then we have a two y sine of u and we have a y sine of u.

00:05:47.520 --> 00:05:50.740
So if we subtract y sine of u from both sides

00:05:50.740 --> 00:05:53.990
we end up with a y sine of u

00:05:53.990 --> 00:05:58.870
y sine of u to subtracting this from both sides,two y minus y sin of u

00:05:58.870 --> 00:05:59.880
it's just gonna be y sine of u

00:05:59.880 --> 00:06:06.820
is equal to y cosine of u.

00:06:06.820 --> 00:06:09.800
so in order for this, in order to have a solution here,

00:06:09.800 --> 00:06:13.380
in order for this to be a [inaudible] statement.

00:06:13.380 --> 00:06:16.440
remember y cannot be equal to zero in order for this

00:06:16.440 --> 00:06:18.590
to end up having a solution

00:06:18.590 --> 00:06:21.630
the coefficient on y have to equal each other.

00:06:21.630 --> 00:06:24.500
sine of u has to be equal to

00:06:24.500 --> 00:06:26.670
sine of u has to be equal to cosine of u.

00:06:26.670 --> 00:06:29.860
so that's one constraint.

00:06:29.860 --> 00:06:35.020
sine of u has to be equal to cosine of u

00:06:35.020 --> 00:06:36.760
And let's think about the unit circle

00:06:36.760 --> 00:06:40.640
and think about how many times are the sine and the cosine

00:06:40.640 --> 00:06:43.810
equal to each other when you're going between zero and three pi.

00:06:43.810 --> 00:06:50.930
So I got the unit circle right over here

00:06:50.930 --> 00:06:54.360
now clearly when we're at forty five degrees

00:06:54.360 --> 00:06:57.480
sine and cosine are equal to each other or forty five degrees

00:06:57.480 --> 00:07:01.300
is the same thing as pi over four

00:07:01.300 --> 00:07:03.220
there they're equal to each other

00:07:03.220 --> 00:07:05.350
you might be tempted to do this over here

00:07:05.350 --> 00:07:07.610
but here the cosine is negative sine is positive.

00:07:07.610 --> 00:07:08.570
so that won't work

00:07:08.570 --> 00:07:11.480
they're both negative over here but they're equal.

00:07:11.480 --> 00:07:13.190
so that's another value

00:07:13.190 --> 00:07:14.940
and then this won't work

00:07:14.940 --> 00:07:16.710
So so far we've traveled to two pi.

00:07:16.710 --> 00:07:19.610
we can go another half, we can go three pi,

00:07:19.610 --> 00:07:21.610
so we can go back to this one again

00:07:21.610 --> 00:07:23.990
so we can go back to this value again.

00:07:23.990 --> 00:07:29.900
so there are one two and then when you go all the way around again

00:07:29.900 --> 00:07:32.310
three values and then we can't go back to this one

00:07:32.310 --> 00:07:33.200
because we can only go -

00:07:33.200 --> 00:07:34.690
Let me just use another color.

00:07:34.690 --> 00:07:38.160
We can only go three pi for u. So we can only go -

00:07:38.160 --> 00:07:41.460
that's two pi and then go another time around.

00:07:41.460 --> 00:07:43.290
that is three pi

00:07:43.290 --> 00:07:46.120
so there's one two three values

00:07:46.120 --> 00:07:52.720
so there's three three possible Us.

00:07:52.720 --> 00:07:56.610
Just from this constraint, just one we use this blue equation

00:07:56.610 --> 00:08:00.270
and this magenta equation. Now let's just make sure -

00:08:00.270 --> 00:08:03.530
Let's just make sure that there aren't any further

00:08:03.530 --> 00:08:06.720
that there aren't any further constraints over here.

00:08:06.720 --> 00:08:08.290
so let's see if we can -

00:08:08.290 --> 00:08:11.380
let's use two of the other equations

00:08:11.380 --> 00:08:14.190
and the one that I would want to use

00:08:14.190 --> 00:08:17.170
that seems like there might be some cancellation.

00:08:17.170 --> 00:08:18.480
Well we can just use -

00:08:18.480 --> 00:08:22.180
We could use either this guy and this guy

00:08:22.180 --> 00:08:25.680
y cosine of u plus two z

00:08:25.680 --> 00:08:27.700
Yeah. Why not? So let's use -

00:08:27.700 --> 00:08:30.730
So as long as we're using all three equation in our constraints.

00:08:30.730 --> 00:08:33.510
we will have kind of properly constrained

00:08:33.510 --> 00:08:35.570
all of the possible solutions.

00:08:35.570 --> 00:08:37.030
and so let's think about this.

00:08:37.030 --> 00:08:39.880
This equals that, which is equal to that, which is equal to that.

00:08:39.880 --> 00:08:48.450
so we can write y cosine of u plus z cosine of u

00:08:48.450 --> 00:08:50.660
is equal to this whole thing over here,

00:08:50.660 --> 00:09:04.510
is equal to y sine of u plus y cosine of u plus two z cosine of u

00:09:04.990 --> 00:09:07.450
and then we have a y cosine of u on both sides.

00:09:07.450 --> 00:09:08.950
Those would cancel out.

00:09:08.950 --> 00:09:13.340
we can subtract a z cosine of u from both sides.

00:09:13.340 --> 00:09:16.510
a z cosine of u from both sides

00:09:16.510 --> 00:09:23.340
and so we would get zero is equal to y sine of u

00:09:23.340 --> 00:09:28.510
plus z cosine - plus z cosine of u

00:09:28.510 --> 00:09:32.800
and this seems like a pretty benign statement.

00:09:32.800 --> 00:09:38.880
y sine of u plus z cosine of u is equal to zero

00:09:38.880 --> 00:09:40.480
let me make sure this.

00:09:40.480 --> 00:09:45.930
y sine of u plus z cosine of u is equal to zero that means that

00:09:45.930 --> 00:09:48.920
this is means that two times this is also going to be zero.

00:09:48.920 --> 00:09:54.330
so this is equal to two y sine of u plus

00:09:54.330 --> 00:09:57.090
two z cosine of u is equal to zero.

00:09:57.090 --> 00:09:58.520
The only reason is I multiply it by two

00:09:58.520 --> 00:10:02.320
because by now this looks identical to this

00:10:02.320 --> 00:10:05.370
so - when you're used this equation and this equation,

00:10:05.370 --> 00:10:08.810
I got a constraint that this expression right over here

00:10:08.810 --> 00:10:13.520
this expression right over here essentially needs to be equal to zero

00:10:13.520 --> 00:10:18.380
that this expression over here essentially needs to be equal to zero

00:10:18.380 --> 00:10:19.500
which is ok

00:10:19.500 --> 00:10:22.750
which is ok because u could make the sine equal zero

00:10:22.750 --> 00:10:24.690
or x can be equal to zero.

00:10:24.700 --> 00:10:27.930
remember, they didn't put any constrains on x

00:10:27.930 --> 00:10:30.910
x can be equal to zero so this is really constraining us.

00:10:30.910 --> 00:10:32.970
x can clearly be equal to zero

00:10:32.970 --> 00:10:36.080
which could make this thing equal to zero.

00:10:36.080 --> 00:10:38.080
so it's not limiting our constraints.

00:10:38.080 --> 00:10:41.110
so now we've used all of the information that's in the problem.

00:10:41.110 --> 00:10:44.570
we've used all three of these surfaces

00:10:44.570 --> 00:10:46.820
essentially to find out if there's an inter -

00:10:46.820 --> 00:10:52.250
what the constraints we have for intersection of the three surfaces

00:10:52.250 --> 00:10:53.830
and the only real constraint is that

00:10:53.830 --> 00:10:56.430
the sine of u has to be equal to the cosine of u

00:10:56.430 --> 00:10:58.210
and there're three possible Us

00:10:58.210 --> 00:11:00.730
between zero and three pi that satisfied that.


Series sum example

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.930
Let s sub k for k is equal
to 1, to all the way to 100,

00:00:04.930 --> 00:00:08.890
denote the sum of the infinite
geometric series whose

00:00:08.890 --> 00:00:12.510
first term is k minus
1 over k factorial.

00:00:12.510 --> 00:00:15.200
And the common
ratio is 1 over k.

00:00:15.200 --> 00:00:20.200
Then the value of 100 squared
over 100 factorial plus the sum

00:00:20.200 --> 00:00:23.180
from k equals 1 to 100
of the absolute value

00:00:23.180 --> 00:00:27.180
of this thing over here, k
squared minus 3k plus 1 times

00:00:27.180 --> 00:00:29.170
s sub k is.

00:00:29.170 --> 00:00:31.840
So this looks like a
pretty daunting thing.

00:00:31.840 --> 00:00:33.380
But I guess a good
place to start

00:00:33.380 --> 00:00:39.490
would just to be find a
simple way to express s sub k.

00:00:39.490 --> 00:00:43.380
So if we were to
write s sub k is

00:00:43.380 --> 00:00:49.140
equal to the first term is
k minus 1 over k factorial.

00:00:49.140 --> 00:00:51.040
And then the common
ratio is 1 over k.

00:00:51.040 --> 00:00:56.230
So it's going to be 1 over k
to the 0 plus 1 over k to the 1

00:00:56.230 --> 00:00:58.800
plus 1 over k squared.

00:00:58.800 --> 00:01:01.730
It's just going to
keep going on forever.

00:01:01.730 --> 00:01:04.569
It is an infinite
geometric series.

00:01:04.569 --> 00:01:06.990
Now there is a very
well-known formula

00:01:06.990 --> 00:01:09.790
for the sum of an
infinite geometric series.

00:01:09.790 --> 00:01:13.200
I enjoy the proof of
it so much that we're

00:01:13.200 --> 00:01:14.230
going to do it again.

00:01:14.230 --> 00:01:15.630
And I've done it
multiple times already,

00:01:15.630 --> 00:01:17.505
but it's such a simple
proof that it's always

00:01:17.505 --> 00:01:19.005
good to start from
basic principles.

00:01:19.005 --> 00:01:20.713
If you're doing this
under time pressure,

00:01:20.713 --> 00:01:22.150
you could just use the formula.

00:01:22.150 --> 00:01:25.280
But if we say, let's
just say some other sum.

00:01:25.280 --> 00:01:29.446
Let's call this thing just s.

00:01:29.446 --> 00:01:35.570
s is equal to 1 over k to
the 0 plus 1 over k to the 1

00:01:35.570 --> 00:01:38.830
plus 1 over k squared all
the way, keep going on

00:01:38.830 --> 00:01:39.940
into infinity.

00:01:39.940 --> 00:01:42.430
Now let's multiply
1 over k times s.

00:01:42.430 --> 00:01:45.220
1 over k times s is just going
to be 1 over k times each

00:01:45.220 --> 00:01:45.860
of these terms.

00:01:45.860 --> 00:01:48.060
When I multiply 1 over
k times this term,

00:01:48.060 --> 00:01:50.700
I'm going to get 1 over
k to the first power.

00:01:50.700 --> 00:01:52.740
When I multiply 1 over
k times this term,

00:01:52.740 --> 00:01:55.800
I'm going to get 1 over
k to the second power all

00:01:55.800 --> 00:01:58.050
the way on to infinity.

00:01:58.050 --> 00:01:59.940
So if I were to
subtract this from that.

00:01:59.940 --> 00:02:02.560
So if I were to subtract that--
so this coefficient is 1.

00:02:02.560 --> 00:02:03.790
So it's 1.

00:02:03.790 --> 00:02:07.550
So we have 1 minus
1 over k times

00:02:07.550 --> 00:02:09.150
s-- I'm just
subtracting that-- would

00:02:09.150 --> 00:02:12.426
be equal to all of this stuff
minus all of this stuff.

00:02:12.426 --> 00:02:13.800
That's going to
cancel with that.

00:02:13.800 --> 00:02:14.710
That's going to
cancel with that.

00:02:14.710 --> 00:02:16.168
We're just going
to keep canceling.

00:02:16.168 --> 00:02:18.340
And we're just going to
have this first term here

00:02:18.340 --> 00:02:19.650
which is to the 0-th power.

00:02:19.650 --> 00:02:23.330
Or 1 over k to the 0 power
is going to be equal to 1.

00:02:23.330 --> 00:02:29.700
Or that this whole sum is going
to be equal to 1 over 1 minus 1

00:02:29.700 --> 00:02:31.070
over k.

00:02:31.070 --> 00:02:33.710
And so we can rewrite s sub k.

00:02:33.710 --> 00:02:35.445
So we could rewrite s sub k.

00:02:35.445 --> 00:02:45.039
s sub k is equal to k minus
1 over k factorial times,

00:02:45.039 --> 00:02:47.580
before I write it up here, let's
see if we can simplify this.

00:02:47.580 --> 00:02:49.288
If I want to simplify
this I can multiply

00:02:49.288 --> 00:02:50.980
the numerator and
the denominator by k.

00:02:50.980 --> 00:02:53.470
If I do that, I have
k on the numerator

00:02:53.470 --> 00:02:57.930
and then the denominator
becomes k minus 1.

00:02:57.930 --> 00:03:04.970
So this right here
is k over k minus 1.

00:03:04.970 --> 00:03:07.040
And these will cancel out.

00:03:07.040 --> 00:03:08.422
And so this will
be equal to-- we

00:03:08.422 --> 00:03:10.130
have to be careful
when we cancel it out.

00:03:10.130 --> 00:03:12.075
Because this kind
of precluded-- well,

00:03:12.075 --> 00:03:13.450
I'll talk about
that in a second.

00:03:13.450 --> 00:03:17.615
So this will be equal
to k over k factorial.

00:03:17.615 --> 00:03:19.730
And what's k over k factorial?

00:03:19.730 --> 00:03:22.640
Remember, this is equal
to k over-- k factorial

00:03:22.640 --> 00:03:30.050
is k times k minus 1 times k
minus 2 all the way down to 1.

00:03:30.050 --> 00:03:32.150
So these k's are
going to cancel out.

00:03:32.150 --> 00:03:36.420
So it's going to be equal to
1 over k minus 1 times k minus

00:03:36.420 --> 00:03:38.700
2 times k minus 3,
all the way down to 1.

00:03:38.700 --> 00:03:46.380
Which is the same thing
as k minus 1 factorial.

00:03:46.380 --> 00:03:50.340
Now we want to make sure that we
get the right value for s sub 1

00:03:50.340 --> 00:03:51.620
and s sub 2.

00:03:51.620 --> 00:03:53.030
So let's just get this.

00:03:53.030 --> 00:03:56.260
So s sub 1, what's that
going to be equal to?

00:03:56.260 --> 00:03:58.995
And 0 factorial is
always a bit of a debate

00:03:58.995 --> 00:03:59.870
that people can have.

00:03:59.870 --> 00:04:00.840
But it becomes a
little bit simpler

00:04:00.840 --> 00:04:02.940
when you look at the
original expression here.

00:04:02.940 --> 00:04:06.170
If you put k is equal to 1
here, this just becomes a 0.

00:04:06.170 --> 00:04:07.900
We lose that because
it cancels later on,

00:04:07.900 --> 00:04:10.120
and that's why I was kind
of warning to be careful.

00:04:10.120 --> 00:04:11.780
But we have 1 minus
1 that's clearly

00:04:11.780 --> 00:04:13.460
going to be 0 regardless
of whether you

00:04:13.460 --> 00:04:15.890
consider 0 factorial be 1 or 0.

00:04:15.890 --> 00:04:19.450
It's clearly, clearly
going to be 0 over here.

00:04:19.450 --> 00:04:22.510
So s sub 1 is going to be 0
times a bunch of other stuff.

00:04:22.510 --> 00:04:25.910
So s sub 1 is going to be 0.

00:04:25.910 --> 00:04:27.850
What's s sub 2 going to be?

00:04:27.850 --> 00:04:31.230
s sub 2 is going to be--
Now we can just use this.

00:04:31.230 --> 00:04:39.500
It's 1 over 2 factorial which
is just going to be equal to 1.

00:04:39.500 --> 00:04:41.200
So with that out of
the way, let's think

00:04:41.200 --> 00:04:43.220
about this sum over here.

00:04:43.220 --> 00:04:44.490
Let's think about this sum.

00:04:44.490 --> 00:04:47.570
We have the sum.

00:04:47.570 --> 00:04:51.440
Actually, before I write it,
it's from k equals 1 to 100.

00:04:51.440 --> 00:04:54.150
But let's just actually
figure out so when k equals 1,

00:04:54.150 --> 00:04:55.570
s sub 1 is 0.

00:04:55.570 --> 00:04:57.830
So this doesn't even
contribute to it.

00:04:57.830 --> 00:05:01.600
When k is equal to
2, we have a 2 here.

00:05:05.540 --> 00:05:07.000
Well, let me just
solve it this way

00:05:07.000 --> 00:05:08.442
since we already
got the 2 there.

00:05:08.442 --> 00:05:09.650
And then we can start from 3.

00:05:09.650 --> 00:05:16.100
When k is equal to 2,
s sub 2 is equal to 1.

00:05:16.100 --> 00:05:19.440
And then we have 2 squared.

00:05:19.440 --> 00:05:22.710
So that's 4 minus 6 plus 1.

00:05:22.710 --> 00:05:25.800
So that's negative 1 times
1 which is negative 1.

00:05:25.800 --> 00:05:28.070
But then we take the
absolute value of it.

00:05:28.070 --> 00:05:32.057
So the value when-- for
the second term here,

00:05:32.057 --> 00:05:33.390
the first term is going to be 0.

00:05:33.390 --> 00:05:37.490
The second term here is
just going to be positive 1.

00:05:37.490 --> 00:05:41.520
So this sum over here is
going to be equal to 0.

00:05:41.520 --> 00:05:45.600
This is when k is
equal to 1 plus 1.

00:05:45.600 --> 00:05:49.170
This is when k is
equal to 2 plus,

00:05:49.170 --> 00:05:51.920
and now we can worry
about the sum from k

00:05:51.920 --> 00:05:55.960
is equal to 3 to 100.

00:05:55.960 --> 00:05:57.840
Notice, it was k
equals 1 to 100.

00:05:57.840 --> 00:05:59.420
But I did the first two terms.

00:05:59.420 --> 00:06:00.410
They were a 0 and a 1.

00:06:00.410 --> 00:06:03.530
So now we have to just
worry about 3 to 100.

00:06:03.530 --> 00:06:05.470
And let's rewrite
this expression

00:06:05.470 --> 00:06:07.480
inside the absolute value sign.

00:06:07.480 --> 00:06:10.440
So inside the
absolute value sign,

00:06:10.440 --> 00:06:13.050
we have this thing
over here times

00:06:13.050 --> 00:06:15.980
s sub k. s sub k
is this over here.

00:06:15.980 --> 00:06:18.410
It's k minus 1 factorial.

00:06:18.410 --> 00:06:21.910
So you have that whole thing
being divided essentially

00:06:21.910 --> 00:06:23.475
by k minus 1 factorial.

00:06:26.370 --> 00:06:27.390
This is the s sub k.

00:06:27.390 --> 00:06:29.310
It's 1 over k minus 1 factorial.

00:06:29.310 --> 00:06:30.970
I'm just putting
in the denominator.

00:06:30.970 --> 00:06:32.650
And you have this thing.

00:06:32.650 --> 00:06:35.500
You have that thing
in the numerator.

00:06:35.500 --> 00:06:38.305
Now my first gut instinct
to see if I could factor

00:06:38.305 --> 00:06:41.030
this, no easy way
to factor this.

00:06:41.030 --> 00:06:43.520
But my second gut
instinct, hopefully,

00:06:43.520 --> 00:06:46.530
is to see is if there's any
way that I could express this

00:06:46.530 --> 00:06:51.140
in terms of k minus 1's
or k minus k's or anything

00:06:51.140 --> 00:06:54.960
like that, that somehow
will make an easy pattern.

00:06:54.960 --> 00:06:58.043
Because what tends to be the
case with these type of-- you

00:06:58.043 --> 00:06:59.550
know you have this crazy sum.

00:06:59.550 --> 00:07:02.039
Clearly you're not going to
be able to do it by hand.

00:07:02.039 --> 00:07:03.330
Things are going to cancel out.

00:07:03.330 --> 00:07:06.380
Normally, you have things
like maybe they oscillate,

00:07:06.380 --> 00:07:12.160
maybe you have stuff like 1
plus 2 minus 2 plus 3 minus 3.

00:07:12.160 --> 00:07:14.030
And so all of the
middle terms cancel out,

00:07:14.030 --> 00:07:15.780
and you just have the first
term and the last term.

00:07:15.780 --> 00:07:18.340
That tends to be the case for
things like this when there's

00:07:18.340 --> 00:07:21.320
not some easy
simplification over here

00:07:21.320 --> 00:07:26.090
or some easy formula like
an infinite geometric sum.

00:07:26.090 --> 00:07:27.460
So let's see if we can do that.

00:07:27.460 --> 00:07:28.950
One way we could express this.

00:07:28.950 --> 00:07:30.700
So let me write this
over here separately.

00:07:30.700 --> 00:07:34.450
k squared minus 3k plus 1.

00:07:34.450 --> 00:07:39.940
We could write this as k
squared minus 2k plus 1 minus k.

00:07:39.940 --> 00:07:42.860
Obviously minus 2k
minus k is negative 3k.

00:07:42.860 --> 00:07:46.090
And the reason why I do that is
this becomes a perfect square.

00:07:46.090 --> 00:07:49.330
This is k minus 1 squared.

00:07:49.330 --> 00:07:52.165
k minus 1 squared minus k.

00:07:52.165 --> 00:07:54.540
And this looks interesting
because it's now expressing it

00:07:54.540 --> 00:07:56.552
in terms of simple
terms, especially when

00:07:56.552 --> 00:07:58.010
you're dealing with
factorials, you

00:07:58.010 --> 00:08:00.310
like dealing with k minus
1's and k's and maybe

00:08:00.310 --> 00:08:01.910
it'll simplify a little bit.

00:08:01.910 --> 00:08:04.080
So I'll write this
expression here

00:08:04.080 --> 00:08:08.200
as k minus 1 squared minus k.

00:08:08.200 --> 00:08:09.890
And let's try to simplify it.

00:08:09.890 --> 00:08:11.640
Let's try to simplify
it even more.

00:08:11.640 --> 00:08:14.502
So this whole thing is
going to become, or at least

00:08:14.502 --> 00:08:16.460
we're just dealing with
the sum part right now,

00:08:16.460 --> 00:08:18.709
we're not really dealing
with the 100 squared over 100

00:08:18.709 --> 00:08:20.650
factorial just yet.

00:08:20.650 --> 00:08:24.990
But this sum is going to
be 1 plus the sum from k

00:08:24.990 --> 00:08:31.580
is equal to 3 to 100 of
the absolute value of-- we

00:08:31.580 --> 00:08:35.784
have k minus 1 squared
over k minus 1 factorial.

00:08:35.784 --> 00:08:37.159
Let me just do it
separately here

00:08:37.159 --> 00:08:39.299
so I don't have to rewrite this
sigma notation over and over.

00:08:39.299 --> 00:08:41.919
This expression right over
here is the exact same thing

00:08:41.919 --> 00:08:50.570
as k minus 1 squared over
k minus 1 factorial minus k

00:08:50.570 --> 00:08:53.240
over k minus 1 factorial.

00:08:53.240 --> 00:08:54.950
It's the exact same
thing, but what's

00:08:54.950 --> 00:08:58.000
interesting about this term,
remember, k minus 1 factorial,

00:08:58.000 --> 00:09:02.420
that's k minus 1
times k minus-- sorry,

00:09:02.420 --> 00:09:04.530
k minus-- let me
write it this way.

00:09:04.530 --> 00:09:11.080
k minus 1 factorial is k
minus 1 times k minus 2

00:09:11.080 --> 00:09:12.650
all the way down to 1.

00:09:12.650 --> 00:09:14.359
That's what this term
is right over here.

00:09:14.359 --> 00:09:15.983
So we have a k minus
1 squared up here.

00:09:15.983 --> 00:09:17.690
We have a k minus 1
in the denominator.

00:09:17.690 --> 00:09:23.130
This will cancel out with one
of the k minus 1's out here.

00:09:23.130 --> 00:09:26.100
This first term over here
simplifies to k minus 1

00:09:26.100 --> 00:09:30.011
in the numerator over-- we
don't have k minus 1 factorial

00:09:30.011 --> 00:09:30.510
anymore.

00:09:30.510 --> 00:09:33.050
We now have k minus 2 times
k minus 3 all the way down.

00:09:33.050 --> 00:09:37.840
So now this is k
minus 2 factorial.

00:09:37.840 --> 00:09:43.510
And this is still minus k
over k minus 1 factorial.

00:09:43.510 --> 00:09:45.980
And this looks interesting now.

00:09:45.980 --> 00:09:48.340
This looks pretty interesting.

00:09:48.340 --> 00:09:50.730
It looks like a
pattern might emerge.

00:09:50.730 --> 00:09:52.720
If whatever the k's,
you go in the numerator

00:09:52.720 --> 00:09:55.080
down one, on the
numerator and down two

00:09:55.080 --> 00:09:56.730
on the denominator
for the factorial,

00:09:56.730 --> 00:09:58.086
and then you subtract it.

00:09:58.086 --> 00:09:59.460
But then for the
next k, it looks

00:09:59.460 --> 00:10:01.070
like you'll have some cancel.

00:10:01.070 --> 00:10:03.100
Let's try it out
with some numbers.

00:10:03.100 --> 00:10:04.840
So this is going to be equal to.

00:10:07.830 --> 00:10:09.720
When k is equal to
3, this expression

00:10:09.720 --> 00:10:13.285
is going to be the absolute
value of 2 over 1 factorial.

00:10:13.285 --> 00:10:19.610
2 over 1 factorial minus
3 over 2 factorial.

00:10:24.639 --> 00:10:25.930
So that's when k is equal to 3.

00:10:25.930 --> 00:10:29.550
And to that we're going to
add when k is equal to 4.

00:10:29.550 --> 00:10:32.850
When k is equal to 4, we're
going to have 4 minus 1

00:10:32.850 --> 00:10:34.990
is 3 over 2 factorial.

00:10:34.990 --> 00:10:39.265
3 over 2 factorial minus
4 over 3 factorial.

00:10:42.010 --> 00:10:43.510
I could keep going.

00:10:43.510 --> 00:10:46.450
Let's do it when
k is equal to 5.

00:10:46.450 --> 00:10:55.290
When k is equal to 5, I have 4
over 3 factorial minus 5 over 4

00:10:55.290 --> 00:10:55.790
factorial.

00:10:58.090 --> 00:10:59.340
And I think you see a pattern.

00:10:59.340 --> 00:11:02.060
We're going to keep
doing this all the way

00:11:02.060 --> 00:11:04.420
to what's the
absolute last term?

00:11:04.420 --> 00:11:08.370
The absolute last term
is k is equal to 100.

00:11:08.370 --> 00:11:12.830
So we go all the way
to k is equal to 100.

00:11:12.830 --> 00:11:17.640
At 100, you're going to have
the absolute value of 99

00:11:17.640 --> 00:11:27.806
over 98 factorial minus
100 over 99 factorial.

00:11:27.806 --> 00:11:29.680
Now the pattern you
might-- and of course, we

00:11:29.680 --> 00:11:30.930
have this 1 out front.

00:11:33.629 --> 00:11:35.170
Now what's this
going to simplify to?

00:11:35.170 --> 00:11:36.253
You might see the pattern.

00:11:36.253 --> 00:11:37.914
We're subtracting a
3 over 2 factorial,

00:11:37.914 --> 00:11:39.330
we're adding a 3
over 2 factorial,

00:11:39.330 --> 00:11:42.120
we're subtracting a
4 over 3 factorial,

00:11:42.120 --> 00:11:43.087
then we're adding it.

00:11:43.087 --> 00:11:45.420
And you might say, well, what
about the absolute values?

00:11:45.420 --> 00:11:49.400
But when you look over
here, this the second term

00:11:49.400 --> 00:11:53.790
of these absolute
value signs is always

00:11:53.790 --> 00:11:59.240
going to be a smaller
value than the first term.

00:11:59.240 --> 00:12:03.850
And that's because we have
a larger value over here.

00:12:03.850 --> 00:12:05.415
We only have k
minus 1 factorial.

00:12:05.415 --> 00:12:07.599
And factorials
increase quite quickly.

00:12:07.599 --> 00:12:09.390
We only have k minus
1 factorial over here.

00:12:09.390 --> 00:12:12.080
We only have k minus
2 factorial over here.

00:12:12.080 --> 00:12:14.650
So this is always going
to be less than this.

00:12:14.650 --> 00:12:16.770
This right here is 3/2.

00:12:16.770 --> 00:12:18.190
This right here is 2.

00:12:18.190 --> 00:12:20.640
This right here is, what is it?

00:12:20.640 --> 00:12:23.240
4 over 6.

00:12:23.240 --> 00:12:24.380
So this is only 2/3.

00:12:24.380 --> 00:12:25.130
This is 1 and 1/2.

00:12:25.130 --> 00:12:27.790
So the second term is always
less than the first term.

00:12:27.790 --> 00:12:29.700
So this is going
to just evaluate

00:12:29.700 --> 00:12:31.950
to be the same thing without
the absolute value signs.

00:12:31.950 --> 00:12:35.470
This is always going
to be a positive value.

00:12:35.470 --> 00:12:38.055
So you actually could ignore
the absolute value signs.

00:12:42.130 --> 00:12:45.400
And now it becomes I think
pretty pretty pretty clear how

00:12:45.400 --> 00:12:47.890
the cancellation
is going to happen.

00:12:47.890 --> 00:12:50.950
You can ignore the
absolute value signs.

00:12:50.950 --> 00:12:53.030
And now we have
a situation where

00:12:53.030 --> 00:12:56.240
the second term from this one
for k is equal to 3 cancels

00:12:56.240 --> 00:12:57.820
with the first term
from k equals 4.

00:12:57.820 --> 00:12:59.010
Those cancel.

00:12:59.010 --> 00:13:00.454
Then these two cancel.

00:13:00.454 --> 00:13:02.870
Then this is going to cancel
with the one from k equals 6.

00:13:02.870 --> 00:13:04.140
Going to keep happening.

00:13:04.140 --> 00:13:07.670
This is going to cancel with the
second term from k equals 99.

00:13:07.670 --> 00:13:11.790
And we're just left
with the one out front.

00:13:11.790 --> 00:13:16.770
1 plus this 2 over 1
factorial which is just 2.

00:13:16.770 --> 00:13:20.405
And then we have minus
100 over 99 factorial.

00:13:24.510 --> 00:13:26.817
And remember, we also
had this whole time

00:13:26.817 --> 00:13:29.400
I didn't pay attention to this
100 squared over 100 factorial.

00:13:29.400 --> 00:13:30.550
But I can do it now.

00:13:30.550 --> 00:13:32.460
But what is 100 squared?

00:13:32.460 --> 00:13:35.300
100 squared is 100 times
100 over 100 factorial.

00:13:35.300 --> 00:13:39.980
So that's 100 times 99 times
98 all the way down to 1.

00:13:39.980 --> 00:13:41.010
These cancel out.

00:13:41.010 --> 00:13:45.039
This is the same thing
as 100 over 99 factorial,

00:13:45.039 --> 00:13:46.330
which we're going to add to it.

00:13:46.330 --> 00:13:49.900
So this is 100 over 99
factorial right over here.

00:13:49.900 --> 00:13:51.180
So let's add that.

00:13:51.180 --> 00:13:54.610
We have 100 over 99 factorial.

00:13:54.610 --> 00:13:58.460
Lucky for us, that first term
and this last term cancel out.

00:13:58.460 --> 00:14:03.550
And we are just left with
1 plus 2 is equal to 3.

00:14:03.550 --> 00:14:06.500
So that fairly daunting
thing, the main trick

00:14:06.500 --> 00:14:09.120
was trying to see how to
simplify this in a way

00:14:09.120 --> 00:14:10.750
that maybe a pattern
would emerge.

00:14:10.750 --> 00:14:13.240
And that's really the
hard part of this problem.

00:14:13.240 --> 00:14:17.790
But we get this entire
sum is equal to 3.