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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition) > Unit 7

Lesson 1: Sequences review- Sequences intro
- Worked example: sequence explicit formula
- Worked example: sequence recursive formula
- Sequences review
- Geometric sequence review
- Extending geometric sequences
- Extend geometric sequences
- Extend geometric sequences: negatives & fractions
- Using explicit formulas of geometric sequences
- Using recursive formulas of geometric sequences
- Use geometric sequence formulas

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# Geometric sequence review

In a geometric sequence, the ratio between consecutive terms is always the same. Learn more about Geometric sequences and see some examples. Created by Sal Khan.

## Want to join the conversation?

- how do you make this easier to understand(3 votes)
- My general strategy for making any of the topics easier to understand is to watch the video a half dozen times and use pause to make sure I fully understand each step.

Is there something specific that is tripping you up?(15 votes)

- What is the best way to understand sigma in solving sequences?(2 votes)
- Sigma means sum. So when you have the sigma in front of a sequence definition, it means put a plus sign between each element and add them all up.(4 votes)

- what is the difference between 'series' and 'sequence' ?(2 votes)
- what is the difference between Arthimatic series and arthemitic sequence?(2 votes)
- A sequence is a list of terms.

A series is the sum of a sequence, that is, the sum of the list of terms.

So, sequence: Sn = { 1, 3, 5, 7, 9 }

Series: ΣSn = 1 + 3 + 5 + 7 + 9(2 votes)

- So how would I write an equation for the example at2:21?(1 vote)
- Since the sequence is not being summed, then it can't be equated to anything so you can't write an equation (in the strictest sense of the term) for the example.

If you are merely trying to notate the sequence then:`5•(1/7)^n for n = 0 to ∞`

(3 votes)

- What is the difference between a geometric sequence and an exponential function?

Video about exponential function: https://www.khanacademy.org/math/algebra/introduction-to-exponential-functions/exponential-growth-and-decay/v/exponential-growth-functions(1 vote)- They are quite similar, since in both a geometric sequence and an exponential function, the growth or decay factor is constant. The main difference is that the domain of an exponential function is all real numbers, while the domain of a geometric sequence (when the sequence is thought of as a function) is the positive integers.

Have a blessed, wonderful day!(3 votes)

- How do you identify whether it is a geometric or arithmetic sequence?(1 vote)
- Geometric sequences change by a common ratio, arithmetic sequences change by a common difference. The common ration is determined by the division of the consecutive values in the sequence or series; the common difference, as you may guess by the name, is determined by subtracting the consecutive values in the sequence.(2 votes)

- So what exactly is the difference between sequences and series? Are series just the sum of the terms of the sequence? So would the series of a geometric sequence be a/(1-r)? Where r is the ratio and a is the first time?(1 vote)
- A sequence is a list of numbers. A series is a sum of a list of numbers. When an infinite series converges, we may be able to provide a number or expression to which it converges, but we don't say that number or expression "is" the series. Note that the geometric series converges only for |r|<1.(2 votes)

- so can i say that a geometric sequence is also an arithmatic sequence as multiplication is an arithmatic operation as well?(1 vote)
- so can i say that a geometric sequence is also an arithmatic sequence since multiplication is also an arithmatic operation?(1 vote)

## Video transcript

- [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.