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### Course: Algebra (all content) > Unit 18

Lesson 3: Geometric sequences- Intro to geometric sequences
- Intro to geometric sequences (advanced)
- Extend geometric sequences
- Use geometric sequence formulas
- Explicit & recursive formulas for geometric sequences
- Explicit formulas for geometric sequences
- Converting recursive & explicit forms of geometric sequences
- Recursive formulas for geometric sequences
- Sequences word problems

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# Intro to geometric sequences (advanced)

Sal introduces geometric sequences and gives a few examples. Notation used in this video is relatively advanced. Created by Sal Khan.

## Want to join the conversation?

- How do you find the common ratio?(8 votes)
- You divide the 2nd term by the first term.(15 votes)

- When are geometric sequences used in real-life situations?(6 votes)
- After sequences you will learn about series, which are just the sums of sequences. We can use them to approximate complicated function values using simpler polynomials in numerical analysis, which is all about using calculus techniques with computers. Here is the section from Khan's integral calculus section that introduces the concepts.

https://www.khanacademy.org/math/integral-calculus/sequences_series_approx_calc

Some of the algorithms in your calculator and computer might use these techniques to approximate complicated expressions involving cosx, sinx, e^x etc. So right there is a real life situation. They are also used a lot in electrical engineering.

Remember, there is no "learn this and then forget it" in math. Everything you learn is preparation for what is coming.

Here is a discussion related to your question:

https://www.reddit.com/r/math/comments/21qwsp/applications_of_power_series/(8 votes)

- When Sal was defining the non-geometric sequence explicitly, he wrote,

a sub n with n=1 to infinity with a sub n = n!

I didn't really get what factorial is?(5 votes)- A factorial is just a product of all integers(positive) less than or equal to 'n'

i.e.

n! = n x (n-1) x (n-2) x..........3x2x1

For example , lets say n=5, so n! i.e. 5! will be

5 x (5-1) x (5-2) x (5-3) x (5-4) = 5x4x3x2x1 = 120

if n=100 then 100! will be 100x99x....x3x2x1(8 votes)

- So, this tells you how to move forward, while using the sequence formula, but how do you go backwards? example: The 10th term in a geometric sequence is 0.78125, and the common ratio is -0.5. Find the first term in this geometric sequence.(5 votes)
- The answer Varun gives is slightly wrong, because of rounding, and also sign!

You divide 0.78125 by (-0.5^9) not a rounded version of that number - -0.5^9 is -0.001953125 but even if you entered all the digits, the dividing by the factored number will always be more accurate!

Such that 0.78125/(-0.5^9) = -400 = a

Therefore the first term is actually 400 if we apply the formula ar^n-1 => -400(-0.5^0) = 400

We can prove it by calculating the sequence up to the tenth term, but for briefness let's confirm that a(r^9) = 0.78125:

10th term = -400(-0.5^9) = 0.78125 exactly.

Whereas if a = 400.641 - then the 10th term would therefore be 400.641(-0.5^9) = -0.782501953125 - which is clearly NOT the correct result.

The correct answer for the first term in that geometric sequence is exactly 400.(2 votes)

- Is geometric sequence an exponential function?(4 votes)
- Yes, and the domain is restricted to positive integers.(4 votes)

- why is the name "geometric"??(4 votes)
- It is quite difficult to explain but here's the easiest explanation (well at least my attempt to do so):

Geometry (in arithmetic) means "the theory of numbers", since r varies according to n, and is therefore not a fixed number, we are here in a theory.(1 vote)

- At5:45shouldn't "a sub n" equal a*n!(4 votes)
- Yes, but a equals 1. "a sub n" is not "a."

I hope this clarifies the video!(1 vote)

- I need a quick reminder: what is a recursive vs explicit? I can't seem to find the video that Sal explained this, and I seem to have neglected to write it down. Thanks!(2 votes)
- Recursive means defining the current term in terms of the previous term. E.g., a(n) = a(n-1)+2.

Explicit means defining the current term in terms of the current index value of the sequence. E.g. a(n) = 3*4^n.

(Those were two different examples; they aren't equivalent sequences)

So if i asked you what is the 145th element of a particular sequence, using the explicit definition will make things a LOT easier. If instead you used the recursive formulation, you'd have to know what the 144th element is, which would require you to know the 143rd element, and so on.....which would take all day.(2 votes)

- If each of the numbers which I'm gonna post below is divided by 2, does that mean that the sequence is geometric? Because I'm sure that's it's not arithmetic! Or maybe it's not one of those two. . . I can't tell:

1.6, 0.8, 0.4, 0.2. . .(2 votes)- Yes, it is a geometric series with a common ratio, r, of 0.5.(2 votes)

- At3:47, what was that other sequence that wasn't geometric, and how do you solve it/put it into explicit form? For instance, my sequence is 1, 3, 6, 10, 15, and 21. How could I solve to find the 100th term?
**Side note**just found out this is a triangular sequence. no idea wat that means...(2 votes)- In general, there's no easy way to do this. But for a lot of cases, taking the sequence of differences or the sequence of quotients is a good way to approach this. In your case, the sequence of differences is 2,3,4,5,6,... From there we can say that aₙ = aₙ₋₁ + n+1 and a₁ = 1 so aₙ = 1+2+3+4+...+n+(n+1) = n(n+1)/2.(1 vote)

## 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.