BAM!
Mr. Tarrou.
In this video we are going to take a look at the factorial symbol or factorial notation.
If n is a positive integer, the notation n factorial, or I like to say NNNNN, is the
product of all positive integers from n down through 1.
The definition as it looks is n! equals n times n-1 times n-2... you know n factorial...
n factorial is equal to n times n-1 times n-2....
And what you are looking at in this part of the definition is it is saying that we can
start with any positive integer, I don't know what it is though, but I am going to start
counting by one.
And then I am going to subtract by one again and just continue to count back by one until
I get to where I know I am going to finish which is 3*2*1.
Zero factorial by definition is equal to 1.
It is by definition.
So don't expect some kind of great proof as too, from me any way, as to how zero factorial
becomes one.
But, by definition for the factorial symbol, zero factorial does indeed equal one.
So we know that zero factorial is equal to one, you know, by definition.
And also though, just as a simpler example before I get to this one here in yellow If
I were to write... let's say...
4 factorial... or [yelling] FOUR.
How do you calculate that by this definition?
You have some value...
some value factorial.
You start at that value and count back by one until you get to 1.
So 4 factorial would be equal to 4 times 3 times 2 times 1.
You just again count backwards until you get to one.
When you are done find the product of all those numbers.
So 4 times 3 is 12.. and then 12 times 2 is 24.
Then you multiply by one and it still 24.
So our first example here is write the first three terms of a sequence whose nth term is
A sub n is equal to 3 to the nth power over n+3 factorial.... [yelling] N+3.
And...
Our book, my PreCalculus, is introducing the factorial notation with the introduction of
sequences.
So that is the first example.
Now we are going to find... let's see... the first three terms.
So I want to let n equal 1, and then 2, and then 3 to find the first three terms of the
sequence.
So A sub 1 is going to be equal to... now if I write A sub 1, I am letting the n value
be one.
So 3 to the first power over... again A sub 1 means n equals 1, so plus 1+3 factorial.
That is going to be equal to 3 over...
well, you know 1 plus 3 is equal to 4 factorial.
And then that is going to be equal to 3 over 4 times 3 times 2 times 1.
So we have, as always.. or usually and not always, but as usual with factorials you will
have some cancellation going on.
And we have 3 on the top of our fraction and a factor of 3 in the denominator.
So that will indeed reduce.
You want to do that to keep the numbers as small as possible.
We have 1 over 4 times 2 which is equal to 8.
So A sub 1 has a value of 1/8.
A sub 2, I was trying to make this all fit but I think I am going to need to erase a
part of this as I go.
A sub 2.
Well A sub n is equal to 3^n over n+3 factorial, so A sub 2 is going to be 3 squared over n+3...
or 2+3 factorial.
I will space this out a little bit because I am never going to get this all to fit.
I am doing three parts of this.
So 3 squared is 9... over 2 plus 3 which is 5 factorial... or 9 over [yelling] 5.
And 9 over 5!, is equal to 9 over 5 factorial is... 5 times 4 times 3 times 2 times 1.
Just to keep the numbers smaller again as i do this in my head, 9 is equal to 3 times
3 and we have a factor of 3 in the denominator.
So this will cancel out leaving us with just a 3 in the numerator.
And A sub 2 is equal to 3 over...
5 times 4 is 20 and then 20 times 2 is equal to 40.
So we have our first A sub n value, A sub 1 is equal to 1/8.
We have A sub 2 which is equal to 3 over 40.
And just to make sure that you have it down, as I erase this why don't you pause the video
and try to work out A sub 3 really quickly.
Then I will do it for you and see if you get the right answer.
A sub 3.
Ok.
A sub 3.
3 to the n, so it will be 3 cubed over 3+3 factorial.
That is going to be 3 cubed... is going to be 3 times 3 is 9... times 3 is 27.
Now over 3 plus 3 which is 6 factorial.
And 6! is 6 times 5 times 4 times 3 times 2 times 1.
Now 27 is 3 times 9, or 3 times 3 times 3.
So let's see here.
We can cancel out the 3 with the 27 and get a 9 in the numerator.
And the 9 has a factor of 3 and so does the factor of 6.
So we can cancel again.
And say...
well ok... 9 divided by 3 is 3.
6 divided by 3 is 2.
I don't think I can reduce any more.
So we have... 27 got cancelled out, then we down from 9...
So we have 3 over...
2 times 5 is 10... 10 times 4 is 40... 40 times 2 is 80.
So A sub 3 is equal to 3 over 80.
Let me double check and yes it is.
We are going to get to the next screen and do two more examples of reducing fractions
with factorials.
Then do another example, the last example, dealing with factorials.
Next example.
24 factorial of 22 factorial.
If you were to put that into your calculator it is very plausible that your calculator
will give you an error and just not do the problem at all.
Or it will give you a question....
Well it might work.
It depends on the type of calculator.
But it is very easy with factorial questions to overload the calculator and have it just
simply not give you an answer.
So these questions with that factorial symbol, especially the ones that are in fraction form,
you want to expand those a little bit by hand... not especially, but all of them...
You want to expand those by hand and just think about the cancellation before you go
and just grab your calculator right away and see if you can't get an answer.
So 24 factorial over 22 factorial.
Well factorials are when you again start multiplication and count backwards to 1.
So 24! is 24 times 23 times 22 times 21 dot dot dot, all the way back until you get to
1.
Right.
Factorial.
So we will do the same thing with the numerator and the denominator. 22 factorial is 22 times
21 dot dot dot all the way back until you get to the factor of 1.
So this is 24 factorial divided by 22 factorial.
And look at what is going to happen.
All of these are factors.
All of these values are being multiplied together.
And when you have a like factor in the top and the bottom of a fraction, they reduce
right.
Like 5 divided by 5 is equal to 1.
So 22 divided by 22 is 1.
21 divided by 21 is one.
And indeed all of these values starting after 22 are going to cancel with the denominator.
And again we can do this because all of these values are being multiplied together.
They are all factors.
Well, what are we left with?
24 times just 23.
And 24 times 23, get that written correctly, is equal to 552.
So, that is how you deal with... or evaluate factorials in a fraction.
Expand them a little bit, you do the cancellation, and then maybe if you need to... use your
calculator at the end to finish figuring it out.
If you put 24 factorial in the calculator and think you are going to get a number, even
if it gives you one it will be in scientific notation and not be very useful for this kind
of question.
Not if you want an exact value answer.
2 times n-1 factorial over n! or just n factorial.
Well this is an algebraic fraction.
So all we are left with, or all we can possible do is simply reduce the fraction.
So we have... let's expand this out like we did our numbers and see what happens.
2...
Oh!
Before I continue on with this a lot of students will want to take this value of 2 and distribute
it through the parenthesis.
Because that is what we have always done when there has been a number in front of a set
of parenthesis.
But these parenthesis are a grouping symbol for the factorial symbol.
So you cannot just...
like here.
We have 2 square root of x.
This is not equal to the square root of 2x.
We cannot just distribute, or multiply, that math function without really thinking about
what is going on.
So we cannot do the same thing with the factorial symbol.
Do not put this 2 through the parenthesis.
We must apply the factorial symbol first, then if we have anything left then we will
multiply the 2 through it.
So we have 2 times n-1... and we are counting backwards, but now it is in general form.
Think of the definition I just erased.
n factorial was n times n-1 times n-2 times n-3 all the way back to the value of 1.
That is because, you know, what do you do when you are making numbers smaller by a value
of 1?
You subtract one.
So the next number here is going to be what? n-1 minus 1 is n-2.
And indeed the pattern of counting backwards by one is going to continue until you get
down to a value of 1.
I don't know what the number is that I am starting with, but I am going to count back
until I get to 1.
And n factorial is n times n-1 times n-2... again just subtracting by one... until I get
back to my value of 1.
Cancel every common factor from the numerator and the denominator.
That is going to be everything from n-1 back.
And 2 times n-1 factorial over n factorial... or 2 times N-1!!!
OVER N!!!... is equal to 2 over n.
Ok.
We have one more example.
It is going to be related to the summation notation (Sigma) and we are done.
BAM!
Last example.
Express the sum using summation notation.
Again, we are studying sequences right now in PreCalculus.
I have already done a video with some of these examples at the end of it, two examples.
This one though incorporates the factorial notation.
So we are going to try and write a summation notation expression.
You have a choice of what your lower limit is.
You have a choice of what your upper...
Well you don't really have a choice for the upper limit.
The upper limit is going to depend on how you start your summation index.
Whether you start at zero, one, two, three, or whatever.
And then we are going to fill up... you know... an answer.
So we can start our k value with anything we like.
And I kind of did not leave myself very much room here.
Let's say k is equal to .... Now I am going to pause a minute and let you try to figure
this out.
But just in case you check work with mine, and there is not necessarily just one right
answer for this problem, but I am going to say that when I write my solution I am going
to start my k value off here as 1, 2, 3, 4, and 5.
So see if you can find a pattern that will turn a 1 into a 1/1.
At the same time turn a 2 into a 1 over 3.
A 3 into a 2 and a 5....
A 4 into 6 and a 7... and a 5 into 24 and 9.
And remember handle the numerator and the denominator separately as their own mini question.
We are looking for a expression that turns a 1 into a 1, 2 into a 1, a 3 into a 2, a
4 into a 6, and a 5 into a 24.
You will be looking for another expression that does the same, except for the denominators.
Go ahead a pause this and try to work that out.
These questions are really the most thought provoking problems that are in these sections
I am currently teaching.
You do need to struggle with them a little bit to get those skills down and be able to
do this on a quiz.
Ok.
We are going to start with our lower index value of 1.
And you might notice the pattern in the denominator quicker than the numerator.
Because what is going on in the denominator is we are simply adding by 2.
So I am about to.... in my next video actually... is going to be about the arithmetic sequence.
And an arithmetic sequence is like this when you are constantly adding by 2's.
So you might be able to figure out the denominator expression by using your knowledge of arithmetic
sequences.
But if you are following this along with my PreCalculus class, you have not learned that
yet.
So how do you turn a 1 into a 2... excuse me... a 1 into a 1, a 2 into a 3, a 3 into
a 5, and 4 into a 7, or a 5 into a 9?
Well, if I double these numbers 1 times 2 is 2... 2 times 2 is 4... 3 times 2 is 6...
and 4 times 2 is 8... and 5 times 2 is 10.
So if I take my k values and I double them all I get a product that is just one bigger
than what I have in my denominator.
So my denominator is going to be 2k-1.
When you think you have a pattern figured out, make sure you check it for at least three
of the terms if not more.
So 2 times 1 is 2, and 2-1 is 1.
2 times 2 if 4, and 4-1 is 3.
2 times 3 is 6, and 6-1 is 5.
2 times 4 is 8, and 8-1 is 7.
Ok.
It seems like it is working.
So these are distracting me down here.
Now I need an expression that will work for the numerator.
And I still am going to use these same k values because now I have it set for the denominator.
Well, this pattern until this video may have been brand new to you, but now look...
From 1 to 1 we are multiplying by 1.
From 1 to 2 we are multiplying by 2.
From 2 to 6 we are multiplying by 3.
And from 6 to 24 we are multiplying by 4.
So we are multiplying through the numerators, and we keep multiplying by a number that is
one larger.
So that is a factorial pattern.
I multiply by 1, I multiply by 2, I multiply by 3, I multiply by 4.
That is the reverse of the factorial pattern.
So something must be going on here that is factorial.
Now 1 factorial is 1.
That is good.
But 2 factorial is not equal to 1, it is equal to 2.
And 3 factorial is equal to 6.
And 4 factorial is equal to not 6 but 24.
So we are just one step away from the factorial pattern working perfectly.
The problem we have, or why we have the same product value of 1 twice in a row is because
zero factorial is 1.
1 factorial is 1.
2 factorial is 2.
3 factorial is 6.
Our numerator is going to be whatever k, and we subtract one, it will work out.
Our upper limit is going to be 5.
And again, just to double check this put that in the same color.... k-1 factorial...
So 1-1 is zero, and zero factorial is 1.
2-1 is 1, and 1 factorial is 1.
3-1 is 2, and 2 factorial is 2.
4-1 is 3, and 3 factorial... that is 3 times 2 times 1 is 6.
I am Mr. Tarrou.
That is my last example.
BAM!!!
Go Do Your Homework:)