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In this lesson, students will learn how to identify and describe arithmetic sequences. They will also learn how to find the equation of an arithmetic sequence in order to find the value of any given term.
Chapters:
00:00 Definition Of A Sequence
01:12 What's An Arithmetic Sequence?
05:02 Example of An arithmetic sequence
07:39 How to find the term of an arithmetic sequence?
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0:00
hang up so now we're going to talk about
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sequences and equations right now this
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chapter here is one of those chapters
0:07
where you really have to just accept
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what I'm saying without trying to like
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why are we doing this like that kind of
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stuff we need to kind of really like
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back off of that because some of this
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stuff is really abstract right you know
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what the word abstract means it's like
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it's like it's like all right whoa whoa
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kind of like wow so so I need you to
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really focus right
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so now a sequence by definition is an
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ordered list of numbers right there a
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list of numbers that are ordered there a
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specific order to it it's not just
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random right so a sequence is an ordered
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list unordered list of numbers and each
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number right is called a term and pay
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attention come on you that ATT right so
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basically by definition a sequence is an
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ordered list of numbers and each number
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is called term when the difference
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between two consecutive terms is the
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same the sequence is called arithmetic
1:06
and the difference is called the common
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difference what's all that mean we're
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going to find out right so here for
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example I have what you call a sequence
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is it ordered or is it
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random how do you
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know right it's 6 7 89 what's happening
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to this sequence here you're adding one
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every time right so now here's the thing
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these numbers here right see 6 7 8 9
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they are called terms right t e r m
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terms right these numbers are called
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terms right these are the terms of the
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sequence these are the terms of the
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sequence and
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then
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um I put them in the table here for a
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specific purpose right the term number
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one is what what's term number one in
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this
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sequence six right term number one is
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six does that make sense because six is
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my first term right term number two is
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what seven that's the second term right
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so that's what I'm saying term number
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this n here stands for the term number
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right so the first term is six second
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term is seven third term is 8 fourth
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term is nine and then if I keep going
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the fifth term will be
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what the sixth term will be 11 11 and so
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on and so forth right so now here's the
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thing the difference between two
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consecutive terms is if if that
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difference is the same right right do
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you know what the word consecutive means
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like cons consecutive what does that
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mean one after the other one after the
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other right if I say one what's the
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consective number to one two to two is
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what three right so if the difference
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between two consecutive integers is the
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same two consecutive terms is the same
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the the sequence is called arithmetic
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that's what this mean here right so here
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what is the difference between six and
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seven one one what's the difference
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between 7 and 8 One 8 and n one because
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one you have one every time we call this
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the common difference because it is the
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same difference does that make sense
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right so because the difference with two
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consecutive in this um terms is the same
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the sequence is called arithmetic right
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and this SE this difference is called
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common difference right so for example 6
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7 8 9 the common difference is one so
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because that's the same these sequence
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here qualifies as an arithmetic sequence
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right what about 2 5 8 or 11 is that an
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arithmetic sequence 2 5 81 why no why is
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it not yes it is
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why you add three each time right so two
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so after two what's my next term I said
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two 5 8 11 right so the the the the
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difference between the two consecutive
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terms is what three three and three
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because that is three this is called a
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common difference because it is the same
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we're going to say that yes this
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sequence qualifies as an arithmetic
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sequence does that make sense right so I
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gave you an example here as well so we
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have that one right in this sequence
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here we have uh the first term is four
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again these here are the term number
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which is the position right so first
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term is four second term is 8 third term
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is 12 the fourth term is 16 and uh the
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difference between each consecutive term
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is four 4 to 8 the difference is four 8
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to 12 the difference is four and then 12
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to 16 the difference is four so this is
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again an arithmetic sequence does
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everybody understand what an arithmetic
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sequence is if anyone doesn't raise your
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hand and we go over it yeah it's just
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patterns patterns right so what I want
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you all to do now is each one of you
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guys going to give me an example of an
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arithmetic sequence real quick right
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I'll start with you give me one M um one
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one of them
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is
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three
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three 10 okay 3 10 mhm 18 that's not 17
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17 right that's an arithmetic sequence
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because what's the difference between
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three and two 3 and 10 s and then 10 and
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seven
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7 so that's that's qualifies you 10 20
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30 40 10 20 30 40 right Emily give me
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one 5 10 15 20 5 10 15 20 excellent it's
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555 each time right you 4 47 10 and 13
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that's good right because again the
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difference between two consecutive terms
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is always in that case three and 2 4 68
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2 4 68 that's great 222 every time um
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150 250 oh you got to be big
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numbers so it's 5050 every time right
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all right uhhuh
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yeah all right 5 7 9 11 great 222 every
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time
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Kenzie
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68 10 12 all right 6 8 10 and 12 the
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common difference is 2 two2 every time
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right so it's an arithmetic sequence yes
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98 27 36 45 all right good 9
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18 9 no hold on a second 9 what you said
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nine and then 18 18 27 Oh 27 yeah good
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yeah good good oh that was a good
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one 369 12 all right 3 692
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Dexter an example of a sequence that's
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arithmetic
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give me one 26 104 all 26 104 four every
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time yeah um 13
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20 um 13
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26 uh 39 mhm and
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you all right nobody wanted to do that
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negatives
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right why not you can do negatives too
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no fractions no negatives you do one
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with numbers that are huge right so
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those are all arithmetic sequences right
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so those are all examples
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now the next one is going to be the
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challenging portion of it right how do
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you find the term of an arithmetic
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sequence for example if I have right let
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me say I have like s let me give you
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this easy one 7 9 11 and 13 right and
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I'm I'm asking you find the 20th terms
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oh I can talk find the 20th term of this
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sequence 20th term how would you do that
8:01
just keep listing them keep listing them
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right most people are going to keep
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listing them to try to find a 20th term
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right this is first term second term
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third term fourth term and then you
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going to keep listening right yeah good
8:13
luck that's going to be a long day for
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you what if I say find the million term
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so what would you do you going to list
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them all day
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long for the if you look at the 48 plus
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it's always times four so I could just
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do tore you can do something like that
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right and add them up at some point you
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can you can kind of figure it out you
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can still play with it like you say you
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can kind of like but there's a way to do
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it by finding the equation an equation
8:40
that's going to help you find that term
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right so we're going to learn how to do
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that right for example so here I say
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write an equation that describes the
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sequence 7 10 13 16 then find the 15th
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term of that sequence so this what you
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need to memorize you're going to
9:00
memorize this formula the book teaches a
9:02
way that I don't like the way the book
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does it is a little confusing and I
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remember last year and then the year
9:08
before that someone sit like we don't
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like we don't like this so I went and
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did some research and I find who likes
9:16
formulas oh don't say anything who likes
9:20
formulas Pap mou who likes who likes
9:23
okay let me let me let me re ask my my
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my my question who likes things that
9:28
okay this this is how I'm going to do
9:30
every time all right who likes that well
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okay this is the formula I'm going to
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use and you use that formula every time
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who likes that everybody right because
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it doesn't require a lot of what brain
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power you just know all right I'm going
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to use this every time if I just plug
9:45
stuffing it's going to be easy for me
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yeah my brain has too much juice in it
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so so you like to exercise it well guess
9:52
what when you start getting older that
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the brain begins like okay I don't want
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too
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much when you're young you have a lot of
9:59
juice in there right right
10:01
now right so here's what we're going to
10:03
do yeah you found so I found up to 20
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yeah she went she went and she found the
10:08
15th term by just listing out the terms
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we're not going to use that right so
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we're going to use this equation
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here look at this right so I'm going to
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explain what this equation means right
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and once you understand it I need you
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all to pay attention and stop talking
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over there because if you don't get it
10:27
you're not going to get it right I need
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you all to pay attention so this is the
10:32
equation that we're going to use all
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right this is the formula so everybody
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write that it's T1 + n -1 * d right
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everybody write that equation down T1
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and I will explain what each what each
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of these uh element means okay
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T1 who can guess what T1 is I don't know
10:51
term term what which is first term T1 is
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the first term right and D is what uh
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the
10:59
denominator
11:01
no
11:03
oh the difference right the difference
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between that's it T1 is the first term
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T1 is the first term right and then and
11:12
then n is turn number and then n is just
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any number right we don't know what n is
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n is just n and then D is the difference
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right and once you know these two you
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can find an equation for your sequence
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very simple stuff so would it be um so
11:29
what's the equ so be what in this in
11:32
this sequence what would be our first
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ter in this sequence about seven is the
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first term by the equation be equal to
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7+ what n - 1 * D What's
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d i
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mean three three that's it you see how
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easy that was last year they wish I gave
11:51
him that last year we had to go through
11:53
a whole lot of stuff and they were
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pretty annoyed because this is too
11:57
complicated so this year I try to make
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it easy for you right all you got to do
12:01
just plug stuff in you just need to know
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what the first term and what difference
12:06
the difference and then you have your
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sequence you're going to apply this
12:10
formula every time and you're going to
12:11
be fine right now now that we Noti this
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right how do we find the 15th term right
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I'm going all I'm going to do is what
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I'm going to replace n by what 15 15 for
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the 15 and for the 20th term I'm going
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to replace n by what 20 20 that's all
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you do once you know the equation to
12:31
find the term you just have to replace
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the N by whatever number I give you
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right so if I want to find the 20th term
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I'm going to replace n by 20 okay right
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because that's the 20 position that's
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what I'm asking for so the 20th term
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will be what 7 + 20 - 1 * 3 right which
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is basically 7 + this is 19 * 3 what's
12:53
19 wait there I'm not sure wait I
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thought it's 15 wait is it ask 20th oh
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wait is it always going to be umus one
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yes it's 60 - 3 20 - 20 is 19 19 * 3 is
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57 right I thought I'll do so this 20th
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term is 64 is that what you got Nora I
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got 45 the 20th term you see so her
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guess was not that great wa did you do
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the 20th term or the 15th so we're not
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doing distributed property I don't need
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to I just did 20 - 1 is 19 so 19
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* uh 3 is
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57 yeah you well there's no variable
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here cuz you replace the N by 20 right
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you don't need to all right you don't
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need to I did it for the seven 9 11 13
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the space in between that is two so that
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the 2 ter for that is 45 45 all yeah
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that makes sense right so this is how
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you're going to do it now what I want
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you all to do here is I want you to find
13:58
for these sequence right
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look here we have another sequence here
14:02
okay I want you to find the 100 the 100
14:04
term right go ahead and do that for me
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so do that on your own find the 100th
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term of the yeah for this sequence here
14:12
using the same formula again we going to
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start with T1 + n minus1 all you need to
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do is identify T1 which is the first
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term and then the difference and then
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whoever has it you can tell me
14:33
can I have a calat
14:40
yep be yeah it's always going to be the
14:44
same way use this
14:46
formula yeah every time it's the same
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formula that's why I give you a formula
14:50
to make your life a lot easier right
14:52
simless yeah um which one you doing we
14:55
doing this one this sequence here we
14:57
find the 100 time are you done yeah all
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right tell me how you did it yeah I've
15:00
done to so mhm I wrote the formula MH
15:04
and then I changed T1 to 4 okay and then
15:09
I changed n to 100 and D to 4 all right
15:12
100 - 1 * 4 right yeah all right so what
15:16
did you get and I 99 * 4 which is
15:19
396 and you add four four 4 400 all
15:23
right which is 400 so the the 100 time
15:26
is 400 right you see that with the
15:28
formula you don't have to go through a a
15:30
rigorous work of like counting terms
15:33
this is why we use this formula and then
15:35
we can find it by just using the
15:36
equation all right if I ask you for
15:40
the a billion term you can also do it
15:42
because you know the formula you just
15:44
have to replace n by the term number and
15:46
then solve for it okay so I thought this
15:49
was going to be harder
15:50
than you know the other one yeah but
15:53
it's actually this one was easier the
15:55
other hard EAS right it's easier all
15:57
right so now you know what we're going
15:59
to do is give your homework
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