WEBVTT
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PROFESSOR: Welcome
back to recitation.
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In this video, what
I'd like us to do is,
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do a little bit of practice
with sigma notation.
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So this will be just
a few short problems
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to make sure that you're
comfortable with what
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all the pieces in the
sigma notation actually do.
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We're going to start
with two problems here.
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And the first one is going to
be a fill-in-the-blanks type
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of problem.
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And the object
is, I've given you
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a sum on the left-hand
side, and then
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I've given you two
other sums, but I've
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left in each place two blanks,
and I've filled in the rest.
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You have enough information
to fill in the two blanks.
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So what I'd like you
to do in this problem
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is fill in the two blanks
so that the sums are equal.
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And the object is
obviously is to do this
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without writing
out all the terms
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and adding up and
then going backwards.
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So you really want
to try and understand
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what each part of the
sigma notation does.
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The second problem
I'd like you to do
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is a simplification problem.
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There are three finite sums.
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And what I'd like you
to do is combine them
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into a single sum or two sums.
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Do the best you can to get
it as simplified as you can
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without actually
writing out a number,
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but keeping it in some
sort of notation form.
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So the object is just
to combine what you can
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and simplify where you can.
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And then we'll do another
one in a little bit.
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But first let's do these two.
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I'll give you a while to work
on them and then I'll be back.
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OK, welcome back.
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We're going to start
with the first problem.
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So the idea is to
really understand
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what each of these
pieces represents.
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And let's look at the
first sum and make
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sure we understand
what's going on.
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So we have 2 raised to a power.
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And what we do is we
index over k from 1 to 5.
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So we're going to take 2 to the
first, plus 2 to the second,
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all the way up to
2 to the fifth.
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And that's where the sum stops.
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Now in this summation,
k is indexed
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from some number I haven't
told you yet, up to 7.
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And I didn't specify
what power of k we want.
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So there are a couple ways
you can think about this.
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It's maybe easiest to work
from what we have up here.
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We know that the
exponent, last exponent
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we would like on the power
of 2 is a 5 in the end.
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But right now, if we just put
a k here, the power would be 7.
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So what we'd like to do is
make whatever the power is
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up here-- based on that
7-- we'd like that power
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to be 2 less than the number
that we're putting in there.
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So probably we would like
this to be a k minus 2,
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because notice then, the last
number you put in, you get a 5.
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Which corresponds to the last
number you put in here, a 2
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to the fifth.
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Now the last number
here is 2 to the fifth.
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And this now will dictate what
we put in the blank down here.
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Because the first
value of k we wanted
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here, the first term
we wanted in this sum,
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was 2 to the first.
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So the first term
we want in this sum
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is going to be 2 to the first.
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So that means that we
would like k to start at 3.
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Another way to
think about this is
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that we know we want the
same number of values
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that we're summing over.
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So notice that from
1 up to 5, there
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are 5 values we're summing over.
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From 3 up to 7,
there are actually
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5 values we're summing over.
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You might think there are
4, because 7 minus 3 is 4,
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but you actually have
to count: 3, 4, 5, 6, 7.
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You see in fact there
are 5 values there.
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So don't get confused by that.
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The differences are the same.
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5 minus 1 is 4.
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7 minus 3 is 4.
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So that's good.
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We have the same number of
things we're summing up over.
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And the first
terms are the same.
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And then you notice, because
of the way we've written,
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it actually is going
to be exactly equal.
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You could expand and check.
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but these are going
to be equal sums.
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Now the third one, I was
a little trickier, maybe.
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I pulled out a factor of 2.
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And so now what
we've done is we've
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taken one of the 2's that
was in all of those terms
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and we pulled it out.
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Right?
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So what do we have here?
00:04:09.680 --> 00:04:13.270
Well we still have 2 to the k.
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But what does this
actually equal?
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To make it easier
on myself, I'm going
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to rewrite this in another way.
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If I pull the 2 back in,
I get a 2 to the k plus 1.
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So now what I've done is I've
given you this 2 pulled out.
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What it's actually doing is it's
changing the exponent value.
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But again, what do we want
the exponents to run over?
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We want them to start,
this exponent to start at 1
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and to end at 5.
00:04:38.940 --> 00:04:41.470
So to get it to start
at 1 and end at 5,
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I need k to be 0 to
start, and finish at 4.
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And that will be sufficient.
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Now again, let's just make sure
that this makes sense to us.
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If k is 0, I get
2 to the 0 here.
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But when I multiply
by a 2 in front,
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the first term is
2 to the first.
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Which is the first term here.
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Let's just check
one more to make
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sure we feel good about it.
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When k equals 1, I get a 2
to the first here, times 2.
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So that's a 2 squared.
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That's the second term
in this sum is 2 squared.
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The second term in this sum
is when I put in k equals 2,
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I get a 2 squared.
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So we see in fact that I've
chosen these values in blue.
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Now these three sums
are actually equal.
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If you're still
nervous about it,
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maybe you can expand the
sums and look at them
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and notice that they are
indeed going to work.
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Now what I'd like us to do is
work on simplifying a problem.
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And if you'll notice, I've put
in three sums, the values here,
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two of them are from 1 to 100,
one of them is from 45 to 100.
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And the three different
things that I'm
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summing: n cubed minus n
squared, n cubed minus n
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squared minus n, and then n.
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And I wanted us to simplify
this as much as we could.
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Now because these
are finite sums
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we can split up over
the terms, as long
00:06:03.660 --> 00:06:05.330
as we keep the right index.
00:06:05.330 --> 00:06:10.670
So let me actually use the
regular chalk for this,
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and I'm going to look at how
I can split up the second term
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to help with the
first and the third.
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So in the second term, notice
I have an n squared and an n
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cubed-- or n cubed
minus n squared here,
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and an n cubed minus
n squared here.
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So what I can do is, I'm
going to look at those terms
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together.
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And then I'm going to look at
the n, the terms-- or summation
00:06:31.470 --> 00:06:33.180
with n and the summation with n.
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And we'll compare them.
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So let me write out what we get.
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We're going to leave the first
one alone for the moment.
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And then I'm going
to subtract off
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this part of that summation.
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And what's left
in that summation
00:06:58.050 --> 00:07:01.200
is every term I
had a minus n also.
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So I'm going to pull that
minus out with this negative.
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And what I'm doing is I'm
taking 45-- n equals 45 to 100
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of these added up.
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And then n equals 45 to
100 of this added up.
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So I end up with another term.
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n equals 45 to 100 of just n.
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So those two terms are
coming from the middle one
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split into two pieces.
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And then the last term,
I just write down.
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So now, it's set up
to go nicely for this
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into a single summation.
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And this into a
single summation.
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And then we'll see if we
can combine them further.
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So if you look here, I have
n equal 1 to a 100 of a sum.
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And then I have n equal 45
to a 100 of the same sum.
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What's that actually mean?
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That means I'm seeing the 45
to 100 thing here, and here.
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And there's a difference.
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Right?
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So I plug in n equals 45 here.
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I get 45 to the third
minus 45 squared.
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I plug in n equals 45
here, I get the same thing.
00:08:11.160 --> 00:08:13.040
And I'm subtracting.
00:08:13.040 --> 00:08:16.250
So what's actually happening
is all the terms that have,
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that show up in both
this sum and this sum
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are being subtracted off.
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What are those terms?
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Those are all the terms
for n equal 45 up to 100.
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Because it's in this
summation and this one
00:08:27.170 --> 00:08:28.570
goes all the way
from 1 to a 100.
00:08:28.570 --> 00:08:31.610
So it certainly
includes 45 to 100.
00:08:31.610 --> 00:08:35.080
So, in fact, you see that all
you wind up with in the end
00:08:35.080 --> 00:08:41.690
is n equals 1 to 44 of
n cubed minus n squared.
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Again why is that?
00:08:43.820 --> 00:08:50.610
This has the 1 through 44 terms
and it has the 45 to 100 terms.
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This has the 45
through 100 terms only.
00:08:53.770 --> 00:08:55.640
So the 45 to 100
terms are in both
00:08:55.640 --> 00:08:57.520
and they're subtracted off.
00:08:57.520 --> 00:09:00.290
So that's one way to think
about why we end up with n
00:09:00.290 --> 00:09:03.079
equals 1 to 44 of this sum.
00:09:03.079 --> 00:09:04.620
And then let's look
what we get here.
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Well in fact, we see it's
exactly the same kind of thing.
00:09:06.995 --> 00:09:08.350
This is 45 to 100.
00:09:08.350 --> 00:09:09.670
This is 1 to 100.
00:09:09.670 --> 00:09:13.560
But notice now that the minus
is on the 1 to 100 part.
00:09:13.560 --> 00:09:21.420
So I'm actually going to get
negative of n equals 1 to 44
00:09:21.420 --> 00:09:23.310
of n.
00:09:23.310 --> 00:09:28.370
Because the 45 terms here,
45 to 100, are in both.
00:09:28.370 --> 00:09:31.980
So the 45 to 100 here, subtract
off the 45 to 100 here.
00:09:31.980 --> 00:09:33.010
Those all go away.
00:09:33.010 --> 00:09:36.880
But I'm still left with
the minus n equals 1 to 44.
00:09:36.880 --> 00:09:38.500
And now I could
simplify this further,
00:09:38.500 --> 00:09:41.620
if I wanted, into a single sum.
00:09:41.620 --> 00:09:47.910
1 to 44 n cubed minus
n squared minus n.
00:09:47.910 --> 00:09:50.400
Why can I do that so easily?
00:09:50.400 --> 00:09:53.337
They're indexing
over the same values.
00:09:53.337 --> 00:09:54.420
That's an important point.
00:09:54.420 --> 00:09:56.211
If this was indexing
over different values,
00:09:56.211 --> 00:09:58.830
I'd have to change this formula
in order to substitute it in.
00:09:58.830 --> 00:10:01.246
But because they're indexing
over exactly the same values,
00:10:01.246 --> 00:10:05.400
I can just take these two pieces
and put them into a single sum.
00:10:05.400 --> 00:10:08.050
So we're going to stop
those two problems now.
00:10:08.050 --> 00:10:10.383
We're going to do one more
summation notation problem.
00:10:10.383 --> 00:10:11.758
So we're going to
come over here.
00:10:11.758 --> 00:10:15.030
And I'm just going
to ask you to write,
00:10:15.030 --> 00:10:18.010
this is a sum of five terms.
00:10:18.010 --> 00:10:21.380
I'm going to ask you to
write this in sigma notation.
00:10:21.380 --> 00:10:24.360
And the main thing-- there will
be multiple ways to do this.
00:10:24.360 --> 00:10:27.040
So you might come up with a
different answer than I do.
00:10:27.040 --> 00:10:29.470
But I'd like you to work
on it for a few minutes.
00:10:29.470 --> 00:10:31.700
And then when you feel
confident, come back
00:10:31.700 --> 00:10:33.680
and I will show you how
I solved the problem.
00:10:42.020 --> 00:10:44.210
OK, welcome back one more time.
00:10:44.210 --> 00:10:47.030
We're going to try and put
this in sigma notation.
00:10:47.030 --> 00:10:49.910
And I have to tell you that when
I look at this kind of problem,
00:10:49.910 --> 00:10:53.150
and I see the same kind of
factor in each of these things,
00:10:53.150 --> 00:10:55.420
I like to make it as simple
on myself as possible.
00:10:55.420 --> 00:10:57.760
I like to pull out
that factor just
00:10:57.760 --> 00:11:00.400
to make sure that I
can simplify this as
00:11:00.400 --> 00:11:02.930
much as possible before
I go into sigma notation.
00:11:02.930 --> 00:11:06.160
So the common factor
to all of these is 1/5.
00:11:06.160 --> 00:11:07.790
I'm going to pull
out a 1/5 before I
00:11:07.790 --> 00:11:09.990
start doing anything else.
00:11:09.990 --> 00:11:12.220
There I get a 1.
00:11:12.220 --> 00:11:18.140
There I get a minus 1/2
plus 1/3 minus 1/4 plus 1/5.
00:11:21.054 --> 00:11:22.470
Now, if you couldn't
do it before,
00:11:22.470 --> 00:11:23.710
you can probably do it now.
00:11:23.710 --> 00:11:26.640
Because now it's
sort of very obvious
00:11:26.640 --> 00:11:28.534
how these terms are changing.
00:11:28.534 --> 00:11:30.450
So we want to see how
these terms are changing
00:11:30.450 --> 00:11:33.630
and how we could index
them in some variable.
00:11:33.630 --> 00:11:37.940
So let's start with the 1/5 and
I'll start with my summation
00:11:37.940 --> 00:11:40.250
and then we'll figure out
what all the pieces are.
00:11:40.250 --> 00:11:43.390
Now obviously the numerator
in this case is fixed at 1--
00:11:43.390 --> 00:11:46.150
and I've got a fraction here,
so the numerator's fixed at 1--
00:11:46.150 --> 00:11:48.920
but the sign is alternating.
00:11:48.920 --> 00:11:50.380
So how do you alternate sign?
00:11:50.380 --> 00:11:54.300
You're going to take negative
1 and raise it to a power.
00:11:54.300 --> 00:11:57.310
Now the power you raise
it to will depend on
00:11:57.310 --> 00:12:00.010
if you want the first term
to be positive or negative,
00:12:00.010 --> 00:12:01.730
and where you start
your summation.
00:12:01.730 --> 00:12:03.880
So there's a lot of
choices you can make.
00:12:03.880 --> 00:12:09.302
But I'm going to start
my summation, we'll say,
00:12:09.302 --> 00:12:12.770
we'll do it in k and
we'll start at k equals 1.
00:12:12.770 --> 00:12:15.190
And then we'll have to figure
everything out from that.
00:12:15.190 --> 00:12:17.460
So I'm going to start my
summation at k equals 1.
00:12:17.460 --> 00:12:20.780
My first term, I want
to be positive 1.
00:12:20.780 --> 00:12:25.210
So I need my power
to be k plus 1.
00:12:25.210 --> 00:12:28.040
Because now my
power here is going
00:12:28.040 --> 00:12:30.575
to be-- when I put in a 1,
I get a 1 plus 1, I get 2.
00:12:30.575 --> 00:12:31.950
Negative one
squared is positive.
00:12:31.950 --> 00:12:33.230
That's that's what I want.
00:12:33.230 --> 00:12:35.940
You might have done k minus 1.
00:12:35.940 --> 00:12:37.760
If you did k minus 1, that's OK.
00:12:37.760 --> 00:12:40.000
Because k minus 1 is
also an even number.
00:12:40.000 --> 00:12:42.430
So when I take negative
1 and I square it,
00:12:42.430 --> 00:12:44.350
I still get a positive number.
00:12:44.350 --> 00:12:46.580
So there are a lot of choices
one can make and still
00:12:46.580 --> 00:12:48.920
be correct on that power.
00:12:48.920 --> 00:12:50.730
And then, I'm counting up.
00:12:50.730 --> 00:12:53.950
Notice the denominator is
increasing just by 1 each time.
00:12:53.950 --> 00:12:58.040
And so it looks like, I could
do just something like over k.
00:12:58.040 --> 00:13:00.000
Now let's check if
that makes sense.
00:13:00.000 --> 00:13:03.350
Well when k is 1, I get
1 in the denominator.
00:13:03.350 --> 00:13:04.480
This is 1 over 1.
00:13:04.480 --> 00:13:06.394
When k is 2, I get 2
in the denominator.
00:13:06.394 --> 00:13:08.060
When k is 3, I get 3
in the denominator.
00:13:08.060 --> 00:13:09.020
So that looks good.
00:13:09.020 --> 00:13:12.360
And now the only question is,
where should I stop this thing?
00:13:12.360 --> 00:13:14.040
So I have my alternating sign.
00:13:14.040 --> 00:13:15.630
My denominator looks right.
00:13:15.630 --> 00:13:17.650
For what value of k
do I want to stop?
00:13:17.650 --> 00:13:20.740
I want to stop when the
denominator equals 5.
00:13:20.740 --> 00:13:23.580
And so I just need
to put a 5 up here.
00:13:23.580 --> 00:13:25.550
And then I'm done.
00:13:25.550 --> 00:13:28.656
Now, if you wanted to
move the 1/5 back in,
00:13:28.656 --> 00:13:29.780
you could actually do that.
00:13:29.780 --> 00:13:36.680
Maybe your solution
looked something--
00:13:36.680 --> 00:13:38.160
I pull the 1/5 back in.
00:13:42.910 --> 00:13:45.450
And I have 5k in there instead.
00:13:45.450 --> 00:13:46.854
Maybe that was your solution.
00:13:46.854 --> 00:13:48.520
But these are ultimately
the same thing.
00:13:48.520 --> 00:13:50.637
Because really this
is just distributing.
00:13:50.637 --> 00:13:51.330
Right?
00:13:51.330 --> 00:13:52.290
This is a big sum.
00:13:52.290 --> 00:13:53.700
I have a 1/5 out in front.
00:13:53.700 --> 00:13:56.140
And so I multiply
every term by 1/5.
00:13:56.140 --> 00:13:58.342
So I just have to put
a 5 in the denominator.
00:13:58.342 --> 00:14:00.300
So you might have had
something more like this.
00:14:00.300 --> 00:14:02.340
That's still correct.
00:14:02.340 --> 00:14:05.730
So just to stress, that
really the sigma notation,
00:14:05.730 --> 00:14:09.572
it's a good tool to understand
how to manipulate easily.
00:14:09.572 --> 00:14:11.030
So there are probably
more problems
00:14:11.030 --> 00:14:13.377
you can find to practice, if
you're nervous about this.
00:14:13.377 --> 00:14:15.960
But I just wanted to give you a
chance to see a couple of them
00:14:15.960 --> 00:14:17.630
and how we work on them.
00:14:17.630 --> 00:14:18.893
That's where I'll stop.