WEBVTT
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Hi, this is Anthony Varela.
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In this video, we're going to
find the sum of an arithmetic
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sequence.
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So we're going to
be adding terms that
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are in an arithmetic sequence.
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We're going to develop a
formula to calculate this sum.
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And then, of course, we're
going to use that formula
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to calculate a sum of
an arithmetic sequence.
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So let's get started by finding
the sum of an arithmetic
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sequence without a formula.
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So we can just do this by
adding these terms concretely.
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So I could say that
2 plus 6 equals 8.
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And then I'll add 10 to that
to get 18, add 14 to get 32,
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add 18 to get 50, and finally
add 22 to get a sum of 72.
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So sum of all of these terms
in this sequence is 72.
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Now that was pretty
easy because I only
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had six terms in my sequence.
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What if I had 600 terms
or something larger
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that I wouldn't really
want to sit there and add
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all these numbers concretely?
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I would want a
shortcut or a formula.
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So we're going to develop
a formula defined the sum
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of any arithmetic sequence.
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And what I'm going
to do is I'm going
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to add the first and
the last terms together.
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So 2 plus 22 equals 24.
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Now I'm going to
work my way inwards.
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So next, 6 plus 18 is 24.
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And finally, 10
plus 14 is also 24.
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So what I've done is I've
paired two terms together
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to get this common sum that
all the pairs add up to 24.
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And then I can just
add up those pairs,
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24 plus 24 plus 24
equals 72, which
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I know from just a moment ago
is the sum to this sequence.
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And this is going to
help me develop a formula
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to find the sum of any
arithmetic sequence.
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So here I have s sub n.
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That means the sum of these
n number of terms here.
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And what this is is it's the
sum of the first and then
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the last terms.
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But I have to multiply
this sum by a value
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because I have 24
here three times.
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So in this case, I'm
multiplying by 3.
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But why 3?
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Well, I have 1, 2,
3, 4, 5, 6 terms,
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and I've paired them together.
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So I have half as many
pairs as I do terms, right?
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So I'm taking my number of
terms n and dividing that by 2.
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And that is going to be
an outside multiplier
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to the sum of the first
term and the last term.
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So this is my formula
for finding the sum
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of an arithmetic sequence.
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So let's go ahead
and use this formula
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to find the sum of a
different sequence.
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So we're going to
pull out our sum.
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And notice, what
I need to know is
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I need to know n, the number
of terms in my sequence.
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I need to know the value
of the first term, a sub 1.
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And I need to know the value
of the nth term, a sub n.
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So taking a look at my sequence,
I have 1, 2, 3, 4, 5, 6 terms
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again.
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So I know that n equals 6.
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I can go ahead and write 6
in for every instance of n.
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Now I need to know the
value of the first term.
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I can see that clearly
enough as negative 4
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and the value of the sixth
term, which I see is 11.
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So now I have all of the
numbers I need to find this sum.
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So first, I'll go ahead
and divide 6 by 2.
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That gives me 3.
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And now I can add
negative 4 and 11.
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That gives me a positive 7.
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So the sum here then is
3 times 7, which is 21.
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And if you'd like, you can
go ahead and add these up
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concretely.
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You would get a sum of 21.
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Well, now let's calculate the
sum of this infinite arithmetic
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sequence.
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So we want to find the
sum of the first 23 terms.
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But notice, I don't
have 23 terms listed.
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So I need to find out what the
value of that 23rd term is.
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So I'm actually
going to bring out
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another formula that's
going to help us
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find the value of any nth term.
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And so this formula here, we
have the value of the nth term
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equals the value
of the first term
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plus some common
difference times n minus 1.
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Now, this common
difference is a number
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that we used to go from
one term to the next.
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So we can see here to get
from 4 to 11, we add 7.
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To get from 11 to 18, we add 7.
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To get from 18 to 25, we add 7.
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So our common
difference here is 7.
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So we're going to
use this formula
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to find the value
of the 23rd term.
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So in this case, n equals 23.
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We're finding the
value of the 23rd term.
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Taking a look at what the value
of our first term is, that's 4.
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And our common difference,
as I mentioned before, was 7.
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So now I have everything
I need then to find
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the value of the 23rd term.
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Well, 23 minus 1 is 22.
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I'll multiply that
by 7 to get 154.
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And finally, I'll add that to 4.
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So my 23rd term
has a value of 158.
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So we're going to go ahead
and write that formula down
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in our notes too if
you ever need to find
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the value of a certain term.
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So now we know
everything we need
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to know to find the sum
of the first 23 terms.
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So let's bring out that formula.
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We know that n equals 23.
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We know that a sub 1, the
value of the first term is 4.
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And we know that the value
of the 23rd term is 158.
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So now we can add 4 and
158 and divide 23 by 2.
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So we get 11.5 times 162.
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And that gives us
a value of 1,863.
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That's the sum of the first
23 terms in this sequence.
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So let's review our
notes for finding the sum
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of an arithmetic sequence.
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To find the sum,
we use this formula
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in which you need to
know the number of terms,
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the value of the first term,
and the value of the last term.
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You add up the
first and last terms
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and multiply that by your
number of terms divided by 2.
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If you don't know then the
value of a certain term,
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you can use this formula--
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that a sub n equals a sub 1 plus
a d, that common difference,
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times n minus 1.
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And here are all the
definitions of the variables
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that we see in our two formulas.
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So thanks for
watching this video
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on finding the sum of
an arithmetic sequence.
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Hope to see you next time.