WEBVTT
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Hi, my name is Anthony Varela.
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And today, we're going to
introduce arithmetic sequences.
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So first I'm going to talk
about what a sequence is.
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Then we'll talk about
an arithmetic sequence.
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And then we'll be
using a formula
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for arithmetic sequences
to solve some problems.
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So what is a sequence?
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Well, here's an example
of what a sequence looks
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like in mathematics,
and it is a set
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of numbers in a specific order.
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So this sequence
happens to be what
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we call a "finite
sequence" because there
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are a limited number of terms.
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There are only five
terms in this sequence.
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We also have what we call
"infinite sequences."
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And this has an unlimited
number of terms.
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So here we notice
that this sequence
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is all of our perfect squares--
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1, 4, 9, 16 25.
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And there are, of
course, many, many, many
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more, 36, 49, and
so on and so forth.
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There are an infinite number
of terms in this sequence.
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So sequences can be finite,
with a limited number of terms,
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or infinite, having an
unlimited number of terms.
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So what's an "arithmetic
sequence" then?
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Well.
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Here's an example of
an arithmetic sequence.
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And an arithmetic
sequence is a set
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of numbers in numerical order
with a common difference
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between each term.
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So you've heard me
say "term" before.
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And a "term" when we're
talking about sequences
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refers to the
place, or the order
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of a number in a sequence, such
as first, second, third, et
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cetera.
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So the number 1 is the
first term in this sequence.
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The number 4 is the second
term in this sequence.
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The number 7 is the third
term in this sequence,
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so on and so forth.
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And then you heard me talk
about "common difference."
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What is "common difference"?
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Well, taking a look at the
values in our arithmetic
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sequence, to get from one term
to the next, I have to add 3.
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So to get from 4 to 7, I add 3.
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To get from 7 to
10, I add three.
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And to get from
10 to 13, I add 3.
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So the common difference
is the numerical distance
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between any two consecutive
terms in an arithmetic
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sequence.
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And it's a constant value.
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You see, in this case, the
common difference is always
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three from one term to the next.
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So with arithmetic sequences,
we have that common difference.
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That's important to know.
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So how can I use then
this common difference
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to find, let's say, the next
two terms in this sequence?
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Well, I would just add 3 to
13, and then add 3 again.
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So the next two terms of the
sequence would be 16 and 19.
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Now, what if I want to find,
for example, the 63rd term
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in this sequence?
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What's the value
of the 63rd term?
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Well, it wouldn't be very
much fun to continuously add
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3 and keep track
of how many times
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I'm doing that until I
get to the 63rd term.
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So we use a formula to do this.
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So here's the formula to
find the value of a term
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in an arithmetic sequence.
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So let's break down
these components.
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a sub n equals the
value of the nth term.
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So that means a sub one would
be the value of the first term
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because that it would
be when n equals 1.
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d is the common difference.
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And n is the term number.
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So first, second,
third, et cetera.
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So we're going to
write that down.
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That's an important
formula to know.
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So let's take a look at
an arithmetic sequence
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and develop the formula
defined the nth term,
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or the value of the nth term.
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So here is our formula.
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a sub n equals a sub
1 plus d multiplied
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by the quantity n minus 1.
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So right off the bat, I'm
going to go ahead and identify
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the first term, that is 7, so
I can make that replacement
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right there.
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Now let's figure out
the common difference.
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So what's the common difference?
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I can choose any two consecutive
terms in my sequence.
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So I'm just going
to choose 11 and 15.
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And I notice that the
common difference then
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is a positive 4 because I have
to add 4 to get from 11 to 15.
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So d equals 4.
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And that's it.
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There is my formula--
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a sub n equals 7 plus
4 times n minus 1.
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Let's do another example,
a different sequence here.
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And it's easy to
identify a sub 1.
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That's just the first
term here in the sequence.
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So that's 19.
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Now let's find the
common difference.
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So, once again, I can choose
any two consecutive terms.
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And to get from 19 to
14, I have to subtract 5.
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So you can see here the common
difference can be negative.
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So our d then is a negative 5.
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So I'm going to write in
19 minus 5 times n minus 1.
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So that is the formula to
find the value of the nth term
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with this particular sequence.
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All right.
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So now let's use this formula.
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So here we're given
the formula to find
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the nth term of a sequence.
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We want to find the 11th term.
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That's what a sub 11 means.
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So what I'm going to do then
is just replace n with 11.
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So a sub 11 equals 3
plus 2 times 11 minus 1.
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Well, that would be
3 plus 2 times 10.
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We can just multiply it by 2.
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So it'd be 3 plus 20.
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So a sub 11 then equals 23.
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Now, what this means is
that for the arithmetic
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sequence described by this
formula, of the 11th term
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is 23.
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Let's use the formula
to find something else.
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Here we want to find n.
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So that would be the
number of the term.
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So this statement
here says that 53
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is the value of the
nth term to a sequence
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and we want to find
out that value for n.
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So this is very much like just
solving a multi-step equation
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that you're probably
familiar with.
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So the first thing we want
to do is distribute that 2
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into both n and negative 1.
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So distributing the 2, I have
53 equals 3 plus 2n minus 2.
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Well, now I have some
like-terms I can combine.
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So 53 equals 2n plus 1.
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Now I'm going to subtract 1
from both sides of my equation
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to have 52 equals 2n.
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And, finally, I divide
both sides of my equation
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by 2 so that n equals 26.
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So this tells me that the
value of the 26th term is 53.
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So let's review our introduction
to arithmetic sequences.
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A sequence is a set of
numbers in a specific order.
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And there can be
finite sequences
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or infinite sequences,
limited or unlimited number
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of terms in that sequence.
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And we talked about
arithmetic sequences
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today, which have
a common difference
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between one term and the next.
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And we introduced a formula to
find the value of the nth term
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in a sequence.
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So thanks for watching this
tutorial on an introduction
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to arithmetic sequences.
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Hope to see you next time.