- [Instructor] I'm going to construct a sequence.
We're going to start with some number.
Let's say I start with the number, a.
And then each successive term of the sequence,
I'm going to multiply the,
to get each successive term of the sequence,
I'm going to muliply the previous term
by some fixed non-zero number,
and I'm going to call that r.
So the next term is going to be a,
is going to be a times r.
And then the term after that,
I'm going to multipy this thing times r.
So it's going to be a,
if you multiply ar times r,
that's going to be ar-squared.
And then if you were going to multiply this term times r,
you would get a times r to the third power.
And you could keep going on and on and on and on.
And this type of sequence or this type of progression
is called a geometric.
Geometric Sequence or Progression.
Geometric Sequence.
You start with some first value.
Let me circle that in a different color,
since I already used the green.
So you start with some first value,
and then to get each successive term,
you multiply by this fixed number.
In this case, this fixed number is r.
And so we call r, our common ratio.
Our common ratio.
Why is it called a common ratio?
Well take any two successive terms.
Take this term and this term,
and divide this term by this term right over here,
ar to the third, divided by ar-squared.
So if you find the ratio between these two things.
Let me rewrite this in the same colors.
So if you took ar to the third power,
and were to divide it by the term before it.
So if you were to divide it by ar-squared,
what are you going to be left with?
Well a divided by a is one,
r to the third divided by r-squared is just going to be r.
And this is true if you divide any term
by the term before it.
Or if you find the ratio between any term
and the term before it,
it's going to be r.
And so that's why it's called a common ratio.
And so let's look at some examples of geometric sequences.
So if I start with the number,
if I start with the number five, so my a is five
and then each time I'm going to multiply,
I'm going to multiply by, I don't know.
Let's say I multiply by 1/7.
So then the next term is going to be five over seven.
What's the next term going to be?
Well, I'm going to multiply this thing times 1/7.
So that's going to be 5/7 times 1/7 is 5/49.
So it's going to be five over seven-squared or 49.
If I were to multiply this times 1/7,
what am I going to get?
I'll just change the notation.
I'll get five times,
I don't actually know in my head
what seven to the third power is,
I guess I could calculate it, 280 plus 63,
let's see, so that would be
let's see seven times 40 is 280,
seven times nine is 63,
so you're going to get to 343, I believe.
Let me see, did I do that right?
Seven times 280,
plus 63 is 343,
and you can just keep going.
So this right over here is an example
of a geometric,
of a geometric sequence.
Started with some first value
and each successive value, I multiplied by 1/7.
1/7 is the common ratio here.
Let me give you another one.
Let's say I have this,
let's say I have three and then let's say I have six,
and let's say I have 12.
Then I have 24,
then I have 48.
Is this a geometric series?
And if it is, what is the common ratio here?
Well you could figure out the common ratio
by just taking any two successive terms and dividing.
Well first, you could try it with two terms.
A 12 divided by six is two.
So we have to multiply by two.
To go from 12 to 24, you multipled by two.
To go from 24 to 48, you have to multiply by two.
To go from three to six, we have to multiply by two.
So you get a fixed common ratio.
For any of these terms, if we multiplied by two
and say multiplied by three,
and so we didn't multiply by the same thing,
then it wouldn't be a geometric sequence anymore.
So this clearly is a geometric sequence,
I forgot to mention that.