BAM!
Mr. Tarrou.
We are going to continue our discussion about Geometric Sequences by talking about the Nth
Partial Sum.
How do you add up n number of terms in a Geometric Sequence?
Well the formula is S sub n, and n again is denoting where you are in the sequence....
If n is equal to 10, then you are at the tenth number... and we are going to find the sum
of the first ten.
That is equal to a sub 1 times one minus r to the n over one minus r.
Our first example is a very very straight forward question.
We have the values of 1, 2, 4, 8, 16, and 32.
This is clearly a Geometric Sequence because all we are doing is multiplying by 2.
So we have an r value equal to 2, and an a sub 1 value equal to 1.
So, to find the partial sum of these first six terms which clearly we could just do in
our head.
But for teaching purposes we like to start with small numbers, plus I am trying to this
mental math as much as possible in front of the camera.
But anyway, we will just check our work.
32...42...48.. 56... 60... 62... and 63.
So when we apply this formula we should be getting the sum of 63 if we do it correctly.
So s sub 6, because we are going to find the sum of these first six terms is equal to a
sub one which is 1 times 1-r and we have a constant ratio of 2, raised to the n power.
We want to find the sum of the first six terms, so n is equal to 6 over 1minus r which again
is 2.
We will have this issue with the second one, make sure that if your r value is negative
you do not cancel with the minus that is in the equation.
You have to apply exponents before you multiply.
When you have two negatives side by side, and you cancel those and make it a plus sign,
you are doing multiplication.
So don't mess up your order of operations.
We have 1 times, this is going to be 1 minus 2 to the sixth.
2, 4, 8, 16, 32, and 64 over 1-2 which is negative one.
This comes out to be negative 63 because 1 minus 64 is -63 over -1.
And we do know that when we divide two negatives we get a positive answer.
So indeed we do, at least at this very simple example, check out that the partial sum formula
does actually work.
We have another example here.
It is a little bit more complicated because of the r value being a fraction.
But we are looking to find the sum of the first 4 terms.
So again you might still want to just do this mentally in your head, but when we start adding
up a lot more than just a few terms you definitely want to apply this short cut formula.
So s sub 4, because we are going to find the sum of the fist four, is equal to a sub 1
which is 27 times 1 minus r....
Notice that I am using the parenthesis when I plug in my r value.
That usually helps students make fewer sign errors.
So it is 27 or a sub 1, times 1 minus r to the nth power... we are going to just find
the sum of the first four... over 1 minus r which is again is negative one-third.
So you can see I am going to have a respectable amount of fraction work here to do and help
you review it as well.
So we are going to have 27 times 1 minus... again I am not going to cancel out those negatives
because I have to do my exponent first..
We have -1 to the fourth which of course is 1, and then 3 to the fourth.
That is 3, 9, 27, and then 27 times 3 is going to be 81.
All over 1 minus.. excuse me.. 1 plus 1/3.
Now here I can do negative times negative is positive because I am not applying some
kind of exponent that I have to do first.
And note, if you had cancelled these out you would have a sign error.
We need a common denominator so we are going to turn the one into 81 over 81.
And we get 27 times 81 - 1 is 80 over our common denominator of 81, and that is all
over... again we need common denominators.
We are going to change that to 3/3.
We get four-thirds.
I think we are going to have some cancellation here.
Let's go ahead and...
We can cancel stuff right now.
27 and 81, how many 27's are in 81?
81 has a factor of 9 in it.
27 has a factor of 9 in it.
81, might not be so obvious, but it has a factor of 3 and so does 27.
So basically 81 divided by 27 reduces down to be 1/3.
Now we are going to take that 80/3, we are also going to flip up the denominator because
when you divide fractions... this is 80 now over 3 divided by 4 over 3.
And you multiply by the reciprocal of the second fraction.
The denominator will be that second fraction.
We get 80/3 times 3/4.
Now obviously the 3's are going to cancel out leaving us values of 1.
And the 4 is going to cancel out.
80 divided by 4 is equal to 20.
Our partial sum comes out to be equal to 20.
So that is end of my second example.
I would like to do one more example like this but one that has some summation notation.
I will be right back.
So here is our last example.
Another one that is going to have a respectable fraction work in it just to make sure that
you get that review that you need.
And this time though our Geometric expression is written in summation notation.
So I want to find the sum of the first four terms of this Geometric sequence.
How do I know that it is four?
Because I am starting to count at one and I am going to four.
If this were 2 and 4, 4 minus 2 is equal to 2 but I have to add one to be three.
Because we start counting as soon as we see the first term and then second, third, and
fourth.
If we had a value of two here it would be 2, 3, and 4.
So you subtract and then add one.
And you don't have to really do any kind of fancy math like for just... you know... a
lower index of one and upper index of four.
But if this were a very large number like maybe 28 and the lower index was 4, 28 minus
4 is 24 and then add one and you would be finding the sum of the first 25 terms.
Ok.
Well, the formula for Partial Sum of a Geometric Sequence is s sub n is equal to A1 times 1-r^n
over 1-r.
And how do I know that this is Geometric?
Because I have an initial amount and I am constantly multiplying receptively by the
value of 3/2.
I know I am doing constant multiplication because I have this variable in the exponent
showing me that is the repetition that I am doing.
Exponents show repetitive multiplication and that is what the geometric setting is again.
So we need a sub 1.
So we are going to let i equal 1.
Let's put it here.
For i equal to 1 we are going to have 8 times 3/2 to the i+1, or 1+1 power.
That going to be 8 times... that is 2, so 3 squared is 9 over 2 squared which is 4.
And 8 divided by 4 is equal to 2.
So 2 times 9 is equal to 18.
So the first term of my Geometric Sequence is equal to 18.
My partial sum...
The sum of my first four terms again going from the lower index of 1 to 4 means that
I will have four terms in this sequence.
S sub 4 is equal to a sub 1 which is 18 times one minus my constant ratio of r, this what
has the... this is the base with the exponent... that variable exponent...
so my r value is 3/2... to the n power... over 1 minus r which is again 3/2.
Ok.
So this is going to come out to be 18 times 1 minus...
3 to the fourth...
again 3, 9, 27, 81... over 2 to the fourth which is 2, 4, 8, 16... over one minus 3/2....
If you want to write that with a common denominator now, that is 2/2.
And 2 minus 3 is negative one over two.
Don't add the denominators, nor do you subtract them.
Just to make sure I don't make any mistakes here, good.
Now we are going to do 1 minus 81 over 16.
So just to make sure i don't run out of space, let's write this 1 with a common denominator
of 16 and make it 16/16.
That is going to give us 18... 16 minus 81... now in my head I would do 81 minus 10 is 71,
and 71 minus 6 is 65.
So this should be negative 65 over 16, over again -1/2.
Again just to make sure I am not making any mistakes here.
Good.
Now let's see.
Let's go ahead and we will keep all of this written in multiplication form.
We are going to flip up that denominator because we are dividing by a fraction, so I want to
show a multiplication of the reciprocal.
We will have...
I had a little interruption there.
So again, we are going to take the denominator and flip it up.
So we are going to have 18/1, we are going to have -65/16, and again we are going to
take denominator and flip it up and have -2/1.
Let's reduce this and multiply it and we will be done.
Two negatives will multiply together to make a positive.
2 and 16, now 2 and 16 will cancel out to be 1 and 8.
Then we can cancel some more because 8 is even and so is 18, so we are going to divide
both of these by 2.
8 divided by 2 is equal to 4, and 18 divided by 2 is equal to 9.
So we have basically...
we have 1 times 9, and then 9 times 65.
Well 9 times 60 is 540, 9 times 5 is 45, so 540 and 45 is equal to 585 over a denominator
of 1 times 1 times 4 which gives us our final answer of 585/4.
I am Mr. Tarrou.
BAM!
Go do your homework:)