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## Geometric sequences

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# Intro to geometric sequences (advanced)

## Video transcript

Let's talk about
geometric sequences, which is a class of sequences
where we start at some number, then each successive number
is the previous number multiplied by the same thing. So what am I talking about? Well let's multiply a times r. And then I'm going to get ar. Let's multiply it times,
but to get the third term, let's multiply the
second term times r. And then what am
I going to have? I'm going to have-- it's a
different shade of yellow-- I'm going to have ar squared. Multiply by r
again, you're going to get ar to the third
power, and you just keep on going like that. And this is, the way
I've denoted this, this is an infinite
geometric sequence. We just keep going on
and on and on and on. And the different
ways we can denote it, we can denote it explicitly. We could say that our
sequence is a sub n starting with the first term going all
the way to infinity, with a sub n equaling-- well,
we see an a here for any term-- is
going to be a times r. And just to be clear,
this right over here, a is the same thing as a times r
to the zeroth power, r to the 0 is just 1. This second term is
ar to the first power. The third term is ar
to the third power. It looks like the
nth term is going to be ar to the n minus 1 power. So ar to the n minus 1. And you could verify it. If you want the second term, you
say a times r to the 2 minus 1, a times r to the first power. It works out. This is defining it explicitly. We could also define
it recursively. We could say a sub n from
n equals 1 to infinity, with a sub 1 being equal to a. That's the base case. a sub 1 is equal to a,
ar to the 0 is just a. Or we could say for n equals
1, and then we could say a-- and I don't even
have to really write that because we're
making it very clear that a sub 1 is equal to a--
and then we could say a sub n is equal to the previous term, a
sub n minus 1, times r, for n is greater than or equal to 2. So this is saying,
look, our first term is going to be a, that right
over there is a, ar to the 0 is just a, and then each
successive term is going to be the previous
term times r, which is exactly what
we did over there. So let's look at some
geometric sequences. So I could have a geometric
sequence like this. I could have a sub n, n is
equal to 1 to infinity with, let's say, a sub n is
equal to, let's say our first term is, I don't know,
let's say it is equal to 20. And then r, the
number that we're multiplying to get
each successive term, let's say it's equal to 1/2. 1/2 to the n minus 1. So what would this sequence
actually look like? Well let's think about it. The first term is 20. If you say, if n
is 1, this is going to be 1/2 to the 0-th power. So it's going to be 1 times 20. So the first term is 20,
and then each time we're multiplying by what? Well here each time
we're multiplying by 1/2. So this could be 20 times
1/2 is 10, 10 times 1/2 is 5, 5 times 1/2 is
2.5-- actually let me just write that as a
fraction, is 5/2, 5/2 times 1/2 is 5/4, and you can just
keep going on and on and on. This is a geometric sequence. Now let me give you
another sequence, and tell me if it is geometric. So let's say we start at 1,
so then I'm going to go to 2, and then I'm going to
go to 6, and then I'm going to go to--
let me see what I want to do-- I want to go to 24. And then I could go to 120,
and I go on and on and on. Is this a geometric sequence? Well let's think
about what's going on. To go from 1 to 2,
I multiplied by 2. To go from 2 to 6,
I multiplied by 3. To go from 6 to 24,
I multiplied by 4. So I'm always multiplying
not by the same amount. You have to multiply by
the same amount in order for it to be a
geometric sequence. Here I'm multiplying it
by a different amount. So this sequence that I just
constructed has the form, I have my first term,
and then my second term is going to be 2 times my first
term, and then my third one is going to be 3 times my second
term, so 3 times 2 times a. My fourth one is 4 times the
third term, so 4 times 3 times 2 times a. And we go on and on and on. So this sequence, which is
not a geometric sequence, we can still define
it explicitly. We could say that
its set or it's the sequence a sub n from n
equals 1 to infinity with a sub n being equal to, let's see
the fourth one is essentially 4 factorial times a. Well, actually, if we
look at this particular, these particular
numbers our a is 1. So this is actually, let
me write this, this is 1, this is 2 times 1, this
is 3 times 2 times 1, this is 4 times 3
times 2 times 1. And so a sub n is just
equal to n factorial. This right over here, which
is not a geometric sequence, describes exactly this
sequence right over here. Just to get some
practice with-- Here we've defined it
explicitly, but we can also define it recursively. We could also say-- do it
in white-- we could also say that a sub n takes
us from n equals 1 to infinity, with a sub 1, or
maybe at a sub 1 is equal to 1. That's our first term. And then each
successive term is going to be equal to the
previous term times n. So the second term is equal
to the previous term times 2. The nth term is going to be
the previous turn times n So this is another valid
way of defining it.