Main content

### 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

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Intro to geometric sequences

Sal introduces

*and their main features, the***geometric sequences***and the***initial term***. Created by Sal Khan.***common ratio**## Want to join the conversation?

- Can anyone explain to me why the geometric sequence follows the basic structure of an exponential function
`[f(x) = a * r^x]`

and why the arithmetic sequence follows the structure of a linear function`[f(x) = mx + b]`

? I get that they are similar concepts but this intrigues me and I would like to understand the reasons behind this similarity. Thanks in advance.(35 votes)- The sequences are in fact linear and exponential functions, but with the domain restricted to positive integers.

The "standard form" of arithmetic sequences, "a(i) = a(1) + d(i-1)", as a function in the xy plane would be "f(x) = m(x-1) + b", where b = a(1) and m = d; which is a shift to the right by one unit of "f(x) = mx + b". This is because the common difference is not added to the first term in this sequence, while in "y = mx + b", m is added to f(x) at x = 1. When the sequence is given as "a(j) = a(1) + dj" (i.e. the common difference is added to the first term) it translates directly to "y = mx + b" with y = 0*x + b when x=0).

Similarly, the 1st term of a geometric sequence is in general independent of the common ratio. So the formula should be "a(i) = a(1)*r(i-1)" (shifted to the right), whereas as a function of real numbers an exponential is "y = (initial value)*r^x"

(Good question).(24 votes)

- Wouldn't the first table be incorrect ( in terms of the jump).

On the second jump, you would do 0.6 times 120 + 120 because 120 is the jump and, 06 times 120 is after the bounce?(4 votes)- Geometric sequences differ from arithmetic sequences. In geometric sequences, to get from one term to another, you multiply, not add. So if the first term is 120, and the "distance" (number to multiply other number by) is 0.6, the second term would be 72, the third would be 43.2, and so on.(14 votes)

- every sequence usually begins with "a1", why we can't begin with "a0" ? I studied the definition of the sequences as functions, that says a sequence is a function with the positive integer as your domain, and the real numbers as your counter domain. Why we can't use the 0 in this domain too?(2 votes)
- You will find that sequences
**don't**always start with a(1). Back here where math is simpler, we do often talk about a(1) as the first item.

It is really a semantics thing. If you mean that the a(1) is your first term, then you cannot have a zero term. Our terms are discrete.

However, you**can**define your first term as a(0) in the same way that in a computer array, the first element is the 0th item. It is usually easier for humans to keep track if the first item is called the n = 1 item.

When we use sequences to match or model actual occurrences, it can get pretty interesting. In the case that Sal is modeling, the first thing that happens is slightly different, so we call it "0"

Whichever way you start numbering, it is always important to check that your formula for the sequence actually ends up with the sequence that you want. If you start at a(1), you will usually need to have your exponent as the expression`n-1`

to match the sequence that you are given.

On the other hand, if your sequence starts with a(0), you will often find your exponent needs to be`n`

in order for your initial value to be correct for building the sequence you want.(15 votes)

- Why did we start at zero bounces instead of 1. In arithmetic we start at 1?(1 vote)
- Yeah, I would say Sal overthought it a bit...

Obviously, there's only*one*jump and the first bounce has to occur*after*that jump.

– – –

So, after her initial (and only) jump Anne reaches a point where the bungee rope will cease extending and start retracting, which I would say counts as her 1st bounce.

At this point the bungee rope measures a length of 120 ft.

Then she is pulled upwards for a couple of seconds until gravity again gets the upper hand and pulls her down to her 2nd bounce, at which point the length of the rope is 60% of the length it measured at the previous bounce, i.e. 120 × 0.6 = 72 ft.

Then at her 3rd bounce, the length of the rope would be

120 × 0.6² = 43.2 ft.

Following the same logic, the length of the rope at Anne's 12th bounce would be 120 × 0.6¹¹ ≈ 0.435 ft.(15 votes)

- Sal, I'm more confused than when I started. This video is convoluted.(7 votes)
- Why is it called "Geometric"? Isn't Geometry about shapes (e.g., triangle, rectangle, etc)?(4 votes)
- It may seem a bit strange, but this can be used to represent shapes. Here's an image to help you understand the "shape" factor of a geometric sequence : https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/GeometricSquares.svg/220px-GeometricSquares.svg.png

Source - https://en.wikipedia.org/wiki/Geometric_series

Hope this helps :)(5 votes)

- when Sal introduced "n", why was it introduced? and does the letter stand for something? I got lost once that "n" was introduced.(4 votes)
- n is the placement number or position of the term you want to find. The first term is always n=1, the second term is n=2, the third term is n=3 and so on. Therefore, this is not the value of the term itself but instead the place it has in the geometric sequence.

Saying "the nth term" means you can calculate the value in position n, allowing you to find any number in the sequence. You use n in the general formula of a geometric sequence and replace it with a number when you want to find the term in a certain position.

For example, if you have the general formula Un = 100 x (2)^n-1, you can use this to find any number in the sequence.

U1 is the first term and U1 = 100 x (2)^1-1 = 100 x (2)^0 = 100 x 1 = 100

U14 is the fourteenth term and U14 = 100 x (2)^14-1 = 100 x (2)^13 = 819200

Hope this helps!(4 votes)

- @1:23We see our first geometric sequence. I know with arithmetic sequences we're always adding, are we always multiplying with geometric sequences?(3 votes)
- We also can divide in a geometric sequence. . .(1 vote)

- At2:10sal multiplies 90 by -1/3. Why is it only mulitplication in geometric sequences but not divison?(3 votes)
- You can represent the division using a fraction. Multiplying by -1/3 is the same as dividing by -3.(5 votes)

- I always remind myself that no matter how hard it is, I Khan always do it.(4 votes)

## Video transcript

In this video I want to
introduce you to the idea of a geometric sequence. And I have a ton of more
advanced videos on the topic, but it's really a good place to
start, just to understand what we're talking about
when someone tells you a geometric sequence. Now a good starting point is
just, what is a sequence? And a sequence is, you
can imagine, just a progression of numbers. So for example, and this isn't
even a geometric series, if I just said 1, 2, 3, 4, 5. This is a sequence of numbers. It's not a geometric sequence,
but it is a sequence. A geometric sequence is a
special progression, or a special sequence, of numbers,
where each successive number is a fixed multiple of
the number before it. Let me explain what
I'm saying. So let's say my first number is
2 and then I multiply 2 by the number 3. So I multiply it
by 3, I get 6. And then I multiply 6 times the
number 3, and I get 18. Then I multiply 18 times the
number 3, and I get 54. And I just keep going
that way. So I just keep multiplying
by the number 3. So I started, if we want to get
some notation here, this is my first term. We'll call it a1 for
my sequence. And each time I'm multiplying
it by a common number, and that number is often called
the common ratio. So in this case, a1 is equal to
2, and my common ratio is equal to 3. So if someone were to tell
you, hey, you've got a geometric sequence. a1 is equal to 90 and your
common ratio is equal to negative 1/3. That means that the first term
of your sequence is 90. The second term is negative
1/3 times 90. Which is what? That's negative 30, right? 1/3 times 90 is 30, and then you
put the negative number. Then the next number is going
to be 1/3 times this. So negative 1/3 times this. 1/3 times 30 is 10. The negatives cancel out,
so you get positive 10. Then the next number is going
to be 10 times negative 1/3, or negative 10/3. And then the next number is
going to be negative 10/3 times negative 1/3 so it's going
to be positive 10/3. And you could just keep going
on with this sequence. So that's what people talk
about when they mean a geometric sequence. I want to make one little
distinction here. This always used to confuse me
because the terms are used very often in the
same context. These are sequences. These are kind of a progression
of numbers. 2, then 6, then 18, 90, then
negative 30, then 10, then negative 10/3. Then, I'm sorry, this is
positive 10/9, right? Negative 1/3 times negative
10/3, negatives cancel out. Right. 10/9. Don't want to make
a mistake here. These are sequences. You might also see the
word a series. And you might even see
a geometric series. A series, the most conventional
use of the word series, means a sum
of a sequence. So for example, this is
a geometric sequence. A geometric series would be 90
plus negative 30, plus 10, plus negative 10/3, plus 10/9. So a general way to view
it is that a series is the sum of a sequence. I just want to make that clear
because that used to confuse me a lot when I first learned
about these things. But anyway, let's go back to
the notion of a geometric sequence, and actually do a word
problem that deals with one of these. So they're telling us that Anne
goes bungee jumping off of a bridge above water. On the initial jump, the cord
stretches by 120 feet. So on a1, our initial
jump, the cord stretches by 120 feet. We could write it this way. We could write, jump, and then
how much the cord stretches. So on the initial jump,
on jump one, the cord stretches 120 feet. Then it says, on the next
bounce, the stretch is 60% of the original jump, and then
each additional bounce stretches the rope 60% of
the previous stretch. So here, the common ratio, where
each successive term in our sequence is going to be
60% of the previous term. Or it's going to be 0.6 times
the previous term. So on the second jump, we're
going to start 60% of that, or 0.6 times 120. Which is equal to what? That's equal to 72. Then on the third jump, we're
going to stretch 0.6 of 72, or 0.6 times this. So it would be 0.6 times
0.6 times 120. Notice, over here, so on the
fourth jump we're going to have 0.6 times 0.6 times
0.6 times 120. 60% of this jump, so every
time we're 60% of the previous jump. So if we wanted to make a
general formula for this, just based on the way we've defined
it right here. So the stretch on the nth
jump, what would it be? So let's see, we start at 120
times 0.6 to the what? To the n minus 1. How did I get this? Let me write this a
little bit here. So this is equal to 0.6,
actually let me write the 120 first. This is equal to 120
times 0.6 to the n minus 1. How did I get that? Well we're defining the first
jump as stretching 120 feet. So when you put n is equal to 1
here, you get 1 minus 1, 0. So you have 0.6 to the
0th power, and you've just got a 1 here. And that's exactly what happened
on the first jump. Then on the second jump, you put
a 2 minus 1, and notice 2 minus 1 is the first
power, and we have exactly one 0.6 here. So I figured it was n minus 1
because when n is 2, we have one 0.6, when n is 3, we have
two 0.6's multiplied by themselves. When n is 4, we have 0.6
to the third power. So whatever n is, we're taking
0.6 to the n minus 1 power, and of course we're multiplying
that times 120. Now and the question they also
ask us, what will be the rope stretch on the 12th bounce? And over here I'm going
to use the calculator. and actually let me correct
this a little bit. This isn't incorrect, but
they're talking about the bounce, and we could call the
jump the zeroth bounce. Let me change that. This isn't wrong, but I think
this is where they're going with the problem. So you can view the initial
stretch as the zeroth bounce. So instead of labeling it jump,
let me label it bounce. So the initial stretch is the
zeroth bounce, then this would be the first bounce, the second bounce, the third bounce. And then our formula becomes
a lot simpler. Because if you said the stretch
on nth bounce, then the formula just becomes 0.6
to the n times 120, right? On the zeroth bounce, that was
our original stretch, you get 0.6 to the 0, that's
1 times 120. On the first bounce, 0.6 to
the 1, one 0.6 right here. 0.6 times the previous
stretch, or the previous bounce. So this has it in terms of
bounces, which I think is what the questioner wants us to do. So what about the 12th bounce? Using this convention
right there. So if we do the 12th bounce,
let's just get our calculator out. We're going to have 120 times
0.6 to the 12th power. And hopefully we'll get order
of operations right, because exponents take precedence over
multiplication, so it'll just take the 0.6 to the
12th power only. And so this is equal
to 0.26 feet. So after your 12th bounce,
she's going to be barely moving. She's going to be moving about
3 inches on that 12th bounce. Well, hopefully you found
this helpful. And I apologize for the slight
divergence here, but I actually think on some level
that's instructive. Because you always have to make
sure that your n matches well with what your
results are. So when I talked about
your first jump, I said, OK this is 1. And then I had 0.6 to
the zeroth power, so I did n minus 1. But then when I relabeled things
in terms of bounces, this was the zeroth bounce. This makes sense that this is
0.6 to the zeroth power. This is the first bounce,
so this would be 0.6 to the first power. Second bounce, 0.6 to
the second power. It made our equation a
little bit simpler. Anyway, hopefully you found
that Interesting.