BAM!
Mr. Tarrou.
We are going to start talking about Geometric Sequences.
I thought I would try some other way of entering the camera for a change.
A sequence in which each term after the first is obtained by multiplying the preceding by
a fixed non-zero constant.
Now I just posted a couple of videos about working with Arithmetic Sequences.
Now we are moving on to Geometric.
So instead of constantly or repetitively adding, now we are repetitively multiplying.
Ok.
So.. common ratio right here.
Ok.
We have got A sub n is equal to A sub 1 times r to the n-1 power.
We have multiplication with the Arithmetic Sequences because multiplication indicates
repeated addition.
Exponents are for repeated multiplication.
Common ratio, as you look as your list of numbers and you try to identify if it is arithmetic
or geometric, go through those values... and this is very simple, we are just multiplying
by 2... but back divide.
2 divided by 1.
4 divided by 2.
8 divided by 4.
Take any term in the sequence, divide it by the previous term, and don't do that just
once... do that a number of times to make sure that you are getting the same ratio over
and over and over again.
Make sure it does not just start off looking like it is geometric but really not.
And don't forget with these sequences your domain value is, the values of n that you
are allowed to plug into your A sub n expression, are just your positive integers.
Now this is not the first time that we have taken an initial value and had a repetitive
multiplication applied to it.
We talked about in our past studies exponential growth and decay.
If a is greater than one, then it is exponential growth.
If a is less than one, then it is exponential decay... or between 0 and 1.
We have also had a couple of questions about finance, whether it is compound interest compounded
yearly or compounded continuously with the pert formula, of course there are other compound
formulas... annuity formulas and what not.
The difference between these two formulas and this formula, again the biggest difference
is these are continuous graphs or the domain is all real numbers.
Now if these are money problems, application problems, you might argue that your domain
is all real numbers greater than or equal to zero.
But these will display a smooth curve, whereas if you remember with sequences with the domain
just positive integers...
If you were to graph the values from a Geometric Sequence, you would just get a bunch of distinct
individual points above your n values of 1, 2, 3, 4, and so on.
You would not want to connect those dots.
Now in this video we are going to just do a few examples, actually four.
I will give you some information and ask you for the general term of that Geometric Sequence,
the A sub n expression if you will or equation.
So we have values of 1, 2, 4, 8, 16, and 32...
Clearly I am just multiplying by 2, but if I am multiplying by 2 repetitively that is
a Geometric Sequence.
So to write out my formula here A sub n is equal to A sub 1, my first term in my sequence,
times r...
Well I am constantly multiplying by 2... and that is to the n-1 power.
Let's not forget why it is n-1 and not just n.
Because we are starting off with the first term.
We started off with n being 1.
And to get the second term where n equals 2, we only have to multiply by the value of
2 once.
To get to the third term, well yes I am on the third term, but I started off with A sub
1 so to get to the third term I only have to multiply twice by the value of 2.
And then the fourth term, A sub 4, I only have to multiply by 2 three times.
So I am multiplying by r n-1 times.
Ok.
That is pretty much done.
There is nothing else to do with this.
This question here, the second example, is almost as easy as well.
Instead of, actually maybe it is even easier.
At least with this I had to figure out my r value by doing ... you know...
A sub n divided by A sub n-1.
A sub n divided by A sub n-1 and make sure that I get the same ratio.
So this one is just boom!
Let's just write it out.
A sub n is equal to -8 times my r value of -1/4 raised to the n-1 power.
My next example here is a recursion or recursive set of formulas.
I have an equation for A sub 1 and then A sub n is equal to negative two times A sub
n-1.
So in other words to find any term in the sequence, go the one just before it and then
multiply it by -2.
So we might be able...
I can write of course out the equation for A sub n like right now.
If you can't or you are not really sure what it will be, you can write this list out.
The first number is 7.
That is A1.
To find A2, I am going to go the previous number and multiply it by -2 which gives -14.
I talked...
I have a video about recursive formulas so I don't really want to break this down super
detailed.
To find A sub 3... to find A3 I am going to take A2 and multiply it by -2.
Negative fourteen multiplied by negative two is twenty-eight, and then so on.
I am multiplying by negative 2, and I do have an A1 value of 7.
So A sub n is equal to A1 which is 7 times -2 raised to the n-1 power.
Let's get one more example out of the way.
And maybe two more examples and call this video done, then move on to partial sums.
Ok.
Two more examples that are quite a bit more difficult.
Here is a sequence of values and I want... you know... for you to identify what type
of sequence it is.
Is it arithmetic, geometric, or neither.
And then come up with an expression if it is one of those two special types.
Ok.
Why is it impossible for this list of values to be arithmetic?
Well, the reason why is...?
We go from positive, to negative, to positive, to negative, and then so on.
So the only way your signs are going to flip flop back and forth like that is if you are
multiplying by a negative value.
So no way this could be an arithmetic sequence.
Now how about being Geometric?
I cannot just look at this and like my previous example just go... oh...
I am just multiplying by 2.
So what we are going to have to do is back divide a couple of these and make sure that
we get the same answer a couple of times.
We have got -16 over 27 divided by 8 over 9.
Now you know you can't really divide fractions.
You multiply by the reciprocal of the second fraction.
So I am going to erase this and say instead of dividing by, or writing dividing by 8/9,
I am going to multiply by 9/8.
And the 8 and the...
Let me pick a different color here.
The 8 and the 16 are going to cancel out to be negative two.
The 9 and the 27 are going to cancel out to be 1/3.
We have a ratio, and r value, that is equal to -2/3.
That is not enough to prove to me that this is a geometric sequence even though I know
it is.
Let me just check one more time.
You might want to do it two more times just to make sure with your work.
So let's take these two and divide them.
Make sure that we get the r value of -2/3 again.
We are going to take... let's write this a little bit differently.
32/81 divided by -16/27.
Any time you have a fraction over a fraction, I wrote my initial example sideways, you are
going to have to flip up that bottom fraction.
That is the fraction that you are dividing by and we are going to want to multiply by
the reciprocal.
So we are going to multiply by 27/16.
A little bit sloppy there.
Let's see.
What will cancel out between 32/81 multiplied by -27/16?
Well again, 32 and 16 are going to cancel out to be 2/1.
And 27 and 81.
What is 81 divided by 27?
It is going to come out to be negative three.
And I am going to just put the negative down here that is next to this 27.
So 27/81 reduces to be 1/3.
r again is equal to -2/3.
So I checked it a couple of times and I got an r value of -2/3.
If I recognize that I am constantly multiplying by the same value, then I have identified
this as a Geometric Sequence.
So we have An is equal to A1 times my constant r value of -2/3 raised to the n-1 power.
Next example.
Here I am giving you the sixth term in the sequence and I am telling you that the r value
is equal to 1/2.
Well if I am going to write out my general term, my formula for the Geometric Sequence,
then I need to know what A sub 1 is and I already know what r is.
Ok.
This is Geometric but I don't know what A sub 1 is.
However to get down the line I am multiplying by 1/2.
So if I want to go in the opposite direction, get up the line and back to A sub 1 I could
just.. you know... the opposite or the reciprocal of multiplying by a 1/2 and multiply by 2
to go backwards through the list of values and get to A1.
Just work out the pattern.
But this might be A sub 16, or something where you would want to do that much work by hand.
So we are going to set up the A sub n formula.
Ok.
A sub n is equal to A sub 1 times r to the n-1 power.
A sub n, now we are going to work with A sub 6 so A sub 6 is equal to 1/8.
I don't know what A sub 1 is.
That is what I am trying to figure out.
My constant ratio is 1/2.
And again I am given A sub 6, so my n value is equal to 6.
And we have 1/8 equals A1 times 1/2 to the fifth power.
1 times 1 times 1 times 1 times 1 is 1.
2 to the fifth power is 2, 4, 6, 8, 16, and 32.
So we are going to multiply both sides by 32.
32 divided by 8 is equal to 4.
Now I have got my first term, I have got r, and that means that I have got my An formula.
Or I have got my answer.
I am Mr. Tarrou.
That is your introduction to Geometric Sequences.
Let's move on to Partial Sums.
BAM!
Go Do Your Homework:)