BAM!
Mr. Tarrou.
In this video we are going to take a look at finding arithmetic means.
It's something that maybe in your textbook of you have been studying arithmetic sequences.
Where you have a list of numbers and those numbers have a common difference between them.
So what an arithmetic mean is, it is the term or terms between any two none consecutive
terms of any arithmetic sequence.
So I have got a pretty simple example here and then we will work through a couple of
real problems.
We have 2, 5, 8, and a couple of blanks which are going to be our arithmetic means, and
then 17.
So we can see here we go from 2 to 5.
That is an addition of 3.
From 5 to 8, that is an addition of 3.
So we have a common difference of 3 and A sub 1 value, we have listed the first term
off here as 2.
And you might need to work with the General Form of an Arithmetic Sequence A sub n is
equal to A sub 1 plus n-1 times d and come up with that general formula for the arithmetic
sequence, or general expression, whatever your textbook and teacher is calling it, to
help you solve the problem.
But I have given us enough information here to just simply say ok, you are adding by 3.
So our two arithmetic means is 8 plus 3 is 11, and 11 plus 3 is 14.
Now just check it, 14 plus 3 is equal to 17.
Ok.
Let's get into a couple of problems that require some actual work.
What are the arithmetic means between 3.2 and 8.4?
I already have it laid out here.
Except that I have got one too many blanks because I mis-wrote in my notes what was going
on here.
So this is take two.
So we have A1 is 3.2, we have three arithmetic means, and then finally our last term that
we are concerned about.. the first term after the missing arithmetic means of 8.4.
Now like the previous example I need to know what the common difference is.
But I don't have any more of this sequence to see what the common difference is.
So we are going to use this A sub n formula to figure that out.
A sub 1 is 3.2 and A sub 5, the last known term in this sequence, is equal to 8.4.
So we are going to do...
well let's just copy this down.
A sub n equals A sub 1 plus n-1 times d.
Well, our last known value is A sub 5 so A5.
Now if I am going to say A sub 5 instead of A sub n, that means that n is equal to 5.
Let's finish plugging in everything that we know.
A sub 5 is 8.4.
A sub 1 is 3.2... now plus 5-1 is equal to 4... then 4d.
Ok so we are going to solve this for d and find our common difference.
We are going to subtract both sides by 3.2.
We get 5.2 is equal to 4d.
Now we are going to divide both sides by 4.
We get d is equal to... let's see here.
5 divided by 4.
There is one 4 in 5.
That would leave me with a remainder of 1.
So 1.2.. or 12 divided by 4 is equal to 3.
So it is 1.3 is our common difference.
We are pretty much done with all the work that we need to answer this question.
3.2... and add 1.3 because it is arithmetic or constant addition.
3.2 plus 1.3 is...
Let's use a different color there...
4.5.
Add by 1.3 again, we get 4.5... 5.5... 5.8.
Add by 1.3 again, 5.8, now 6.8 plus another .3 is 7.1.
And add by 1.3 again just to make sure that we have not made any mistakes.
7.1 plus 1.3 is equal to 8.4.
That is the first example.
The next example I am going to kind of beat you up with a little bit of fractions and
review of how to work with some radicals.
That's because the only thing that makes these questions more difficult is introducing some
fractions, or radicals, or something that basically makes the arithmetic more difficult
itself.
So let's see that second example.
BAM!
Alrighty then.
Our last example.
Write a sequence with three arithmetic terms between square root of two and sixteen.
Now other than a little bit more arithmetic because of the square root of 2 in there,
this example is going to work out just like the last one.
So you may want to pause the video and see if you can get it done before I show you the
solution.
Now I am just going to list these out.
You know.
Leave the blanks in the sequence.
We have got the square root of 2, we have three arithmetic means, and the last term
is 16.
That means that this is going to be a A sub 1, A2, A3, A4, and ok we have A sub 5.
And we need to find the difference, common difference.
That means we are going to use that General Term of an Arithmetic Sequence.
That is A sub n is A1+(n-1)d.
Now A sub n, again the last term we know A sub 5 after all the arithmetic means that
we are missing.
We are going to have A sub 5, or take the 5 and plug it in for n.
That means that this is 5, because again n is just where you are in the sequence.
A sub 5 is 16.
A sub 1 is square root of 2... 5 minus 1 is 4... d.
Subtract both sides by square root of 2.
That is equal to 4d.
We are trying to solve for d so we are going to divide both sides by 4.
There is our common difference.
Now all I have to do is fill in the blanks.
So A sub 2 is equal to A1 which is square root of 2 over 1 plus our common difference
of 16 minus square root of 2 over 4.
Well you cannot add fractions if you do not have common denominators.
We are going to come through here and multiply the top and bottom of A sub 1 by 4.
Now we have... let's see.
We have common denominators, so we have A sub 2 is equal to 16... 4 minus 1, these square
root of 2's are going to act like adding like terms so 4 minus 1 is equal to 3 square root
of 2 over 4.
Now don't be tempted to cancel the 16 with 4.
You cannot cancel terms.
You can only cancel out factors.
So the 16 and the 4 do not cancel out.
If you have done that, you have made a mistake.
So this is the first arithmetic mean that we where looking for.
Now A sub 3 is going to be A2 plus our common difference of 16 minus square root of 2 over
4.
Add like terms we get 16 plus 16 is 32.
3 minus 1 is 2, so plus 2 square root of 2 over 4.
Now these two terms are both even, so I could factor a 2 out and then cancel that factor
of 2 out of the denominator.
But I have to continually adding by 16 minus square root 2 over 4, so I need that common
denominator.
I am not going to reduce it down.
That is A sub 3.
I just have to do this one more time.
A sub 4, because that is my last arithmetic mean I am looking for.
A sub 4 is equal to A sub 3, so 32 plus 2 square root of 2 over 4 plus again our common
difference of 16 minus square root 2 over 4.
We already have again common denominators, so we are going to keep those and get A sub
4 is equal to 32 plus 16...
Now 32 and 10 is 42, and another 6 is 48.
2 minus 1 is 1 so plus square root of 2.
48 plus square root 2 over 4.
That is our last arithmetic mean.
That is the end of our last example.
BAM!
Do do your homework:D