BAM!!! Mr. Tarrou. In this video we are going to take a look at the summation notation.
This greek e symbol here (sigma) which basically just means repetitive addition. This notation
can be found in a lot of formulas related to finance where you are talking about money.
Maybe it is the total loan value or a total income from an investment. You will also see
it in many many probability questions or probability formulas and formulas that we use in statistics.
This is because statistics is basically a math class that is based off of the study
of probabilities. Now in precalculus we are just going into the topic of sequences and
series and that is how we are using this notation. So, we are just going to do a basic introduction
of the notation, show you how it is repeated form of addition, and then do a couple of
examples where I give you the summation notation and we are going to come up with the terms
in a sequence. Then we will give you a sequence of values and then ask for or try to figure
out what the summation notation or expression would be that would give us that pattern.
We have this sigma, it will be to the left of some kind of algebraic expression. We have
this index of i. Now you don't have to use the letter i, you can call it n, k, whatever
letter you like. But, there is some kind of index. I is called the index of summation.
The n, this number at the top of the sigma is the upper limit of the summation. One is
the lower limit of summation. Now I am writing this pretty much as my book has it, however
it is kind of...the one is just here in the general standard form. The lower limit, it
can be some other integer other than one. Just remember that this is the lower limit
and this is the upper limit and we are going to increase, what we count by, we are going
to increase it by one until we get to the upper limit. So, what you are going to see
in the examples here in a minute is that we have an expression here, whatever this lower
limit is will be the first number that we plug into this expression (in this case it
is one), I will plug one in and get a term, and then increase by one and plug in two to
this expression to get the second term, and then I would plug in three to get the third
term, and so on. Just keep plugging numbers in getting bigger and bigger by values one
until until we reach the upper limit of n. That would be the end of the sequence. We
could have an infinity symbol here, but most of the time you are dealing with finite sequences,
so there will be an actual last number in the sequence. Then you will go back and the
greek letter now come in (sigma), the summation symbol comes in, and you are going to take
all of those terms that you found and add them up and find the sum. So, let's get to
some examples. BAM! We have our three examples where we are given summation notation, the
expression, and we are going to work out the sequence...the finite sequence of numerical
values...then do what the summation says which is go through all those values you just found
add them up and you are done. It is just new notation. The mathematics is pretty simple.
We are just going to do a lot of substitution and a lot of arithmetic until we find the
sum. So, we are going to take this expression two n squared minus three. We are going to
start plugging in values of one, we are going to increase by one until we get to the upper
limit of four. So, we are going to have two times one squared minus three. I am going
to wrap this in parenthesis just to highlight each individual substitution. It is not really
going to do anything except...well just that, to highlight each substitution. We are going
to plug in two and get two squared, or two times two squared minus three. Then we are
going to plug in three. Two times three squared minus three. Then finally reach our upper
limit of four, so two times four squared minus three. All we have left to do is a little
bit of arithmetic and find the sum. One squared is one and one times two is two. So we have
two minus three plus two squared is four, and four times two is eight minus three. That
plus three squared is nine. Don't forget to do your exponents first before you multiply.
So, three squared is nine times two is eighteen minus three. Then four squared is sixteen
times two is 32 minus three. Now we have two minus three is negative one. Eight minus three
is five. Eighteen minus three is fifteen. Thirty-two minus three is twenty-nine. I think
by now you could have probably just dropped those parenthesis. But I was rolling:) We
have got five and fifteen is twenty. Twenty and another twenty-nine make 49. I am thinking
of the answer. Let's try that again. Five and fifteen is twenty, another twenty is forty,
now it is forty-nine minus one is equal to 48. That is all you have to do with the summation
notation. Plug in the values, go from the lower limit to the upper limit by just simply
increasing your substitution values by one, and then do a lot of arithmetic. Now here
is a couple of cases. The summation of k factorial over two, or KAAAYYYYY! over two:) And factorials
if you are not familiar with it, I will explain it as I work through the problem. I have a
full lesson about factorials right after this one. I will make one. And we have a lower
limit and an upper limit, and just like the first example we are going to plug those in.
So, we have three factorial over two plus, now our upper limit increasing by one and
I am done already, so four factorial over two. Now again if you have never seen a factorial
symbol, the factorial tells you to take positive integer and (that is what you are allowed
to plug in) you count down by one until you get to one, then you multiply. I guess actually
on a number line, it would be what? Yeah, count down by one. Ok, so three factorial
is three times two times one, over two plus four factorial, which is four times three
times two times one, over two. Cancel anything out just to keep the numbers smaller. Two
divided by two is one. Here again, two divided by two is one. Now we have three times one
is three. We have plus four times three is twelve. Our final sum is fifteen. Let me make
sure that I got that right, yes. I will cover this in my video about factorials, after I
step back in the screen here. I will cover this in my factorial video, but just as a
verbal note, with these factorials zero factorial by definition is equal to one incase you happen
to see that in your summation problems. Now this last example here, we have the summation
of four with the index values of two to five. We are going to find the summation of four
using the n values two to five. You might be going well this example and this example
gave me variables to plug in these index values. I had a place to plug in my one, two, three,
and four. I had a place to plug in three and four in four k. This this is a constant. This
value of four is never going to change. The expression is four, it is a constant. What
am I supposed to do with two, three, four, and five? Well, that tells you how many terms
that you are going to have. Let's use the index value of two. But, where am I going
to plug it into? No where. But the expression, the value of this expression is going to be
equal to four when n equals two. Well, now n is going to equal three. What is the value
of this expression? It is still four. Now the n value is going to be four. I am starting
to loose place of where I was here. N value of two, n value of three, n value of four,
and then finally an n value of five. So I have still got four index values of two, three,
four, and five. We are attempting to plug into the expression, but the expression is
a constant value of four so its value will not change. But, it is still indicating that
we have four of those fours so our sum is going to be four, eight, twelve, and sixteen.
I am Mr. Tarrou...BAM!!! That is how your work with summations. So now that we know
how to use summation notation, let's try going in reverse. Here is a sequence of numbers,
a finite sequence. It goes from one-fourth to ten over thirteen. Express that sum using
summation notation. Summarize this with a single expression instead of a long series
of additions. Well, when you set up sequence notation you are looking for...going to have
a summation symbol and we are going to have some kind of index down here. We have a choice
as what we want to use as our lower and our upper limit. All we need to then is come up
with some kind of expression that will make these values based on some kind of index number.
We can start this with any value that we like. We could say that n starts at zero then one,
two, and then so on figuring out what that last value is for our last term. How can I
make a zero into a one, a one into two, a two into a three, and an n into a ten. Scratch
that last on there. That is just an upper limit. Then zero to four, one to five, two
to six. Now this is a first example and it is pretty simple one. You could very easily
come up with an expression, actually let's go ahead and do that. We will just do three
examples instead of two. If I want my lower limit to be zero, or my lower index to be
zero, then to get from zero to one, from one to two, from two to three, let's see... I
am taking my values of n and I am simply adding them by one. So, my expression would be n
plus one and that will make all of my numerators out of these n values going from zero to n.
My denominators. How do I make a zero a four, or one to a five, or two into a six? Oh, all
of these denominators are four units bigger than my index value. So, my denominator i
just n plus four. Now, what is the last n? What is the upper limit that I am going to
use in the summation symbol? Well ok, this is my numerator and I need my numerator of
n+1 to equal ten. So, if I take this and I subtract by one then I need my n value to
equal nine. Just to double check that this is going to be my upper limit, so just to
double check that let's see... Zero plus one is one, one plus one is two, two plus one
is three, and if I let nine be my last number nine plus one is ten and nine plus four is
equal to thirteen. So, that is is a correct answer. Now you might have been going , "Mr.
Tarrou if you had just started your n value at one, then this expression could have been...this
is one equals one, two equals two, etc, so my numerator would just be n. My denominator,
well now they are just three bigger. My lower limit I have started at one. My index and
my numerator are a perfect match, so not only am I starting at one but my upper limit must
be ten. So there is two possible answers for the summation notation. Now if you just watched
my lesson about my introduction to sequences and series, when you write an A sub n expression,
when you are doing a bunch of this when we get into our videos for arithmetic series
and geometric series, when you write an A sub n expression for sequence or series of
values the n value must start at one. Now I am not talking A sub n notation, I am talking
about sigma notation which gives me flexibility for my lower limit value. That is probably
why the textbooks will default to using the letter i for the index variable on summation
notation. But half the time I write an i it looks like a one, so maybe I should have picked
a letter like k instead of n to show some separation between A sub n notation that I
used. Just be aware that when you are given a list of values, terms, and you are asked
for an A sub n expression/equation you do need to use your counting values, to need
to remember that your domain is the set of all positive integers. But, with summation
notation you do have a choice for that lower limit. I didn't want to confuse you there.
We have a slightly more complicated version over here. Let's go ahead and use a different
letter for the index. Let's say that we have a summation notation and we are going to let
k be our index value. We are looking for a lower limit. We are looking for an upper limit.
And of course, we are going to fill in our expression here. Let's just default here to
k values of one, two, three, four, and five. There is no dot dot dot with this one. We
just went right to the end. Do you see any pattern between these k values and the numerator?
They are all squared so our numerator is going to be, I don't know why I made my box so small,
but my numerator is going to be k squared. I mentally was checking my answer there. My
lower index is equal to one. With these summation problems where you are trying to write an
expression to match up a series or sequence of numbers, make sure that check more than
the first and second term because you can sometimes, and very commonly, get a pattern
that works a couple of times and then starts to fail. So don't check just one or two of
your terms to make sure that you have come up with a proper expression. Now, how do you
turn a one into a three, a two into a five, a three into a seven, a four into nine, and
a five into eleven. This is a little bit tricky, so I would like to pause here and see if you
can find a pattern that matches all of these. This is two bigger, three bigger, four bigger,
five bigger, six bigger. So, there is not just a simple addition. How about, what is
one times two? Two times two. Three times two. Four times two. Five times two. If we
double all of these value we are one unit away from this denominator. So, my denominator
is going to be 2k. Then it was all just one shy, so how about 2k plus one. Five squared
is twenty-five so our upper limit is five. There is two examples of being given a sequence
and being asked to find the summation notation. I am Mr. Tarrou. BAM! Go Do Your Homework.