Bam! Mr. Tarrou!
in this video, in the next few videos, we are going to be discussing sequences, and then ultimately series
we will be discussing, let see here.. my notes show recursion formulas, some nation notations
defining for you the factorial symbol if you needed
if you haven't seen that before
and then in later videos moving into arithmetic sequences and series,
and geometric sequences and series
but before it, now let get this basic idea of sequence out of the way
an infinite sequence a(s) of n is an function where domain is the set of all positive integers
that is, we are talking about the values of n
despite new notation, its actually fairly simple topic
the notations are all new though, so it could be confusing
the sort of independent variable, if you will, the domain is this value of n
we will see that how it works with the examples
and the domain is the set of all positive integers
the function values, which is gonna be expressed as a(s) of n notations
its kind of like f(x), only is a notation for a sequence
the function values, or terms, of the sequences are represented by a(s) of 1, a(s) of 2, a(s) of 3
and here is n = 1, you are putting it into an expression, and you are getting your a(s) of 1 value,
now you plug-in a value of 2, getting your a(s) of 2 value and so on, until to a(s) of n
and there is a little dot dot dot at the end, because we are defining an infinite sequence
now if there is an ending point, if i want to cover up those dot..dot..dot..
and say that the last term in the sequence is a(s) of n, well, the you are discussing..
um.. excuse me, where you looking at is a finite sequence
so that is the definition,
that is the difference between a infinite and finite sequence
let, get some examples out of the way, shall we ?
to over to examples.
a(s) of n is equals to negative 3 n plus 5, and a(s) n is equals to negative 2 to the nth power
over two to the n plus one.
and the direction says, write the first term of each sequence, and then graph it.
so, what that means is, i want to know what a(s) of 1 is, a(s) of 2, a(s) of 3, a(s) of 4 and a(s) of 5
well, this means that n=1,
n=2, n=3, n=4, and n=5
and we are just gonna plug this values in of 1 through 5 into this expression
and remember, n is the defined as a positive integer
so the lowest value that n can take on is 1
so we have -3 times 1 plus 5,
and -3 times 2 plus 5
and -3 time 3 plus 5, and so on
so what does that mean?
that means that we have got negative 3 times 1 is negative 3,
and the negative 3 plus 5 is equals to 2.
so a(s) 1 is equals to 2.
now we are gonna plug in the value 2, -3 times 2 is -6, and -6 plus 5 is -1.
now lets look n=3, -3 time 3...
let me say let n=.... lets let n=3, make sure sure i say that right, -3 time +3 is -9, and -9 plus 5 is -4
now lets let n=4, and find a(s) of 4, -3 times 4 is -12, and -12 plus 5 is -7
and you might be noticing our values are going down by consisted value of 3
and that because of it is -3 times n, its a linear function if you will
but just for practice, again let be n=5, to find a(s) of 5, -3 times 5 is -15, and -15 plus 5 is equals to -10
lets get this dotted lines out of the way, it looks like i'm writing 17! okay, so boom, boom, boom and boom!
alright! now...
as far as graphing this, lets move this line out of the way
remember what you just copied in the previous screen?
where it said hat the domain of a sequence is set of all positive integers ?
when you are looking at a graph, you know, we are used to graphing smooth and continuous graphs
like parabolas and lines, are smooth and continuous
well, if look at my values here, i'm gonna graph this
and instead calling this my x-axis, i'm gonna call this my n-axis,
because n in sequence is the value that you plug-in in, so its your independent variable.
x and y we're plugging in n and getting
out a sub n values so i'm going to call
this my a sub and axes so my values i
plugged in we're one through five so 1 2
3 4 5 and I got y values or a sub n
values between 2 and negative 10 so I
method carries us down quite a bit
farther and we got 12 and negative 1
negative 2 3 4 5 6 7 8 9 and negative 10
and let's put negative 5 there so we
have a point on my plug in 1 and got two
so when endless one case of endless
too I plugged in to and got negative one
so when n was too I got negative 1 is an
answer when n was three a sub 3 was
negative for when n was for a support
was negative 7 so four five six seven I
started not standing for it again she
want to try and make sure this is
straight and then finally when a sub
when endless 5a safan was equal to
negative 10 now it's very very tempting
to go oh well I'm making a straight line
but if I were to make a straight line I
would be indicating with my graph that
all values of n were included in the
domain but and what you plug into a
sequence expression by definition is the
positive integer so we are definitely
talking about a set of you know just a
set of discrete points not a continuous
line representing an infinite number of
possibilities or an infinite number of
points or indicating my domain is all
real numbers my domain is only the
positive integers so that's a first
example let me get this out of my way
because I think I'm going to need the
space and let's find for this example
now a sub 1 now a sublime means that N
equals 1 so we got negative 2 to the 1
power over to the first power plus one
else will be negative 2 over 2 plus 1
which is equal to negative two thirds
and now we're going to find a sub 2
which means we got negative 2 squared
over 2 squared plus 1 which now when you
start raising a negative number two
powers when the power is even the
negative is going to cancel away when
the power is odd like it was up in the
first example the negative is going to
stay there for a sub n values are going
to oscillate a lot because of this
negative number in my
run the Seas negative 2 squared is
negative 2 times negative 2 which is
four and I got 2 squared is 4 plus 1 is
5 so we have four fifths we have a sub 3
which is going to be negative 2 to the
third over 2 to the third plus one
that's going to be negative 2 to the
third is negative 2 times negative 2
times negative 2 which is negative 8
over 2 to the third which is 8 plus 1
which is 9 so a sub 3 is equal to
negative 8 over 9 and we're going to
find the first five and I'm just going
to kind of be quite here for a second
and see if you can figure out what a sub
4 and a sub 5 are with the same type of
substitution if you want to pause the
tape and we're going to pause the video
and work on that or I'm just give the
answers a sub for comes out to be 16 /
seven teams scuze me 16 / 17 and a sub 5
ends up coming out to be negative 32 /
33 so let's get this scrap we have
remember this was we just leave some
foot goes here n equals one and equal to
and equal three right here right and
equals 4 & N equals five so our graph
negative point 5 and negative one
positive point 5 and positive one
because all my fractional answers are
between negative 1 and 1 and my end
values again we're between one and five
or one
one through five so one two three four
five this is agony my n axis and my a
sub M axis so an a what N equals 1 a sub
1 was negative two-thirds so when n
equal 1 we were a little bit past
negative point 5 when n was too we got
four fifths so when n was too we got
four fist which is point nine little bit
closer to the one that we work to
negative one when n was three we got
negative eight nights so that's getting
a little bit closer we're just basically
getting closer and closer to either
positive 1 or negative 1 when n is for
we got 16 17 which is very close to one
and that's all that's going to happen is
we're going to keep getting closer and
closer to either 1 or negative 1 is our
Edna use get larger and in five we end
up with negative 32 / 33 so that is how
you find terms in a sequence with your
ace of n expression and also might your
ground make sure if you do graph these
that you do not connect the dots as
tempting as that is because the domain
of a sequence is all positive integers
let's move on to the next topic in this
video recursion formulas the defining
nth term of a sequence as a function of
the previous term now as we move through
this topic we're going to be doing
problems where we're given a list of
numbers a sequence or series and we have
to come up with a general equation that
explains or gives us that sequence of
numbers basically you know notice the
pattern and writing equation now
recursion formulas are much easier to
come up with in the general formula that
I'll be preferring my students give me
and you'll see in these examples why now
recursion for analysis topic we have to
cover they do have its place
but we don't prefer them because well
what it says the end of the term of a
sequence is a function of the previous
term so while it's easier to come up
with recursion formulas when you're
looking at the list of numbers in your
sequence if you want the tenth number in
the sequence you have to do the previous
nine you can't go straight to the tenth
one a general formula a general ace of
that formula we do that for you but
let's just see how these recursion
formulas work a sub 1 equals three and I
want to find or my sequence is defined
as a sub n is equal to three times a sub
n minus 1 the subscript means that I
want you to go to the problem the number
just previous to the one you're trying
to find so we have to be told what the
first number is in the sequence because
we're going to use it to find the next
number in the sequence and we're going
to do this for again values greater than
equal to 2 because we're you know that
when N equals 1 our first value is 3 a
sub 1 is equal to 3 so well a sub 1 is
equal to 3 so a sub 2 is equal to three
times a sub n minus 1 or 2 minus 1 minus
fun well a sub 2 minus 1 is a sub 1 + a
sub 1 is equal 3 so we have three times
a sub 1 which is 3-5 and 9-5 is equal to
4 ok let's move on to the next one a sub
3 whoops that's ugly how about a sub 3
is equal to well 3 times a sub 3 minus 1
minus 5 a sub 3 minus 1 well it's 2 so 3
times a sub 2 which is equal to 4 so 3
times 4 minus 5 3 times 4 is 12 and 12
minus 5 is equal to 7 a sub 4 is equal
to three times a sub 4 minus 1 minus 5
so it's three times well or minus 1 is 3
so a sub 3 is equal to 7 minus 5 and 3
times 7 is 21 and 21-5 is 16 and then
finally a sub 5 i'm going to skip the
notation here asa 5 is 3 times the
previous value in our sequence which is
16-5 3 times 10 is 33 times six is
eighteen thirty and 18 is 48 48 minus 5
is 43 so a sub 1 equals 3 a sub 2 equals
4 a sub 3 equals 7 a sub 4 equals 16 and
a sub 5 is equal to 43 okay if I wanted
a sub 6 I could not go straight from a
sub 1 to a sub 6 I have to work through
these lines so again recursive formulas
our recursion formulas are easy to use
easy to come up with but you know will
not allow you to get to any one
particular point in a sequence you have
to start from the beginning and work
your way to whatever valued light one
more time here we got some yellow chalk
and a sub 1 is equal to 2 a sub 2
is equal to the previous value square so
that's going to be 2 squared plus 1
that's four plus one which is five a sub
3 is equal to the previous value a sub n
minus 1 a sub 2 which is 5 squared plus
one that's 25 plus 1 which is equal to
26 and so on so I'm mr. true that's your
introduction to sequences and the next
video is going to be about the summation
notation how to use it what it does and
its affect our factorial symbols either
sir true hey go to your homework