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There's more terminology.

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[MUSIC]

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What is a geometric progression?
A geometric progression, is a sequence

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with a common ratio between the terms.
We should see an example.

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Maybe the sequence starts 3, then 6, 12,
24,

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48, 96 and it keeps on going.
And looks like the

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general rule for this sequence is a sub n
equals 3 times 2 to the n.

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Why is that a geometric progression?

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Well, there's a common ration of 2 between
each of these terms.

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To get from 3 to 6, I have to multiply by
2.

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To get from six to 12, I multiply by 2.
To get from 12 to 24, I multiply by 2.

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To get from 24 to 48, I multiply by 2.
Alright?

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That's the common ratio between all the
terms in this sequence, it's 2.

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We can write down, a general formula for a
geometric progression.

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So I can write a sub n, equals the first
term,

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A sub 0, times the common ratio R to the
nth power.

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In this particular example, A sub 0, the
first term is 3.

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And the common ratio is 2.

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Here's a question, why are these things
even called geometric progressions?

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Well in a geometric progression, each term
is the geometric mean of it's neighbors.

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Okay, but what is a geometric mean?

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Well, the geometric mean of two numbers,
of a

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and b, is defined to be the square root.

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>> Of A times B.

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>> Why is a geometric mean, called
geometric

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at all?
What's geometric about it?

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>> Well, here's on geometric story you
could tell yourself.

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You could build a rectangle, one of who's
sides

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is A, and the other side has length B.

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Then this rectangle has area AB.

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I'm going to build a square.

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And I want to build a square, whose area
is also ab.

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What's its side length?

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Well the side length will be the square
root of ab.

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So this is some kind of geometric sense,
in which an average

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of a and b might deserve to be the square
root of ab.

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A geometric average.

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So the deal with geometric progressions,
is that

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each term is the geometric mean of its
neighbors.

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So let's see that in our original example:
3, 6, 12, 24, and so on.

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The claim is that in a geometric

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progression, each term is the geometric
mean of

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it's neighbors.
Let's see that here.

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What's the geometric mean of 3 and 12.

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Well, it's the square root of 3 times 12,
that's the square root

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of 36, that's 6 so, yeah, 6 is the
geometric mean of it's neighbors.

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Let's try to get em 12.

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What's the geometric mean of 6 and 24?

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Well, that's the square root of 6 times
24.

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6 times 24 is 144.
And the square root of

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144 is 12.

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So, yeah, 12 is a geometric mean of 6 and
24.

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The limit of a geometric progression,

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depends very strongly on that common
ratio.

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Well in our example here, what's the limit
as n approaches infinity of a sub n?

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It's infinity, I can make a sub n as big
as I like, provided I choose n big enough.

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What if the common

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ratio were a third?
Here's an example of a geometric

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progression, with common ratio a third.
1, a 3rd,

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a 9th, a 27th, an 81st, and so on.

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What's the limit.

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Of a sub n in this case as n approaches
infinity.

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Well that's really

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the limit as n approaches infinity of 1/3
to the nth power because that's a formula

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for the nth term in this sequence.
Well, that limit is 0, right?

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By making n big enough, I can make a sub n
as close to 0 as I like.

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Other interesting things can happen, too.

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You should think about what happens, when
that common ratio is negative.

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