If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Sequences intro

Sequences are ordered lists of numbers (called "terms"), like 2,5,8. Some sequences follow a specific pattern that can be used to extend them indefinitely. For example, 2,5,8 follows the pattern "add 3," and now we can continue the sequence. Sequences can have formulas that tell us how to find any term in the sequence. For example, 2,5,8,... can be represented by the formula 2+3(n-1). Created by Sal Khan.

Want to join the conversation?

  • So is a sequence basically just a function where the input is limited to positive integers?
    (180 votes)
    Default Khan Academy avatar avatar for user
    • hopper jumping style avatar for user Ironfire_
      so to sum up what they said, in a sequence you can start with any positive or negative number and then add, subtract, multiply, and/or divide that number (k) and it will repettitively do so until you have specified it to stop. so no it's not always positive, but it can be
      (22 votes)
  • aqualine ultimate style avatar for user ahamed.zoha
    What is the difference between finite and infinite sequence, as they both have similar functions?
    (16 votes)
    Default Khan Academy avatar avatar for user
    • leaf grey style avatar for user Cameron Conley
      In a finite sequence, there are a limited number of values for k. For example, the first finite sequence that Sal lists has values of k from 1 to 4. Because there are four values of k, the sequence only contains 4 numbers and is therefore finite.
      Infinite sequences, on the other hand, contain an unlimited number of values for k. The first example of an infinite sequence that Sal lists has values of k from 1 to ∞. Because there are infinite values of k, the sequence contains infinite numbers and is therefore infinite.
      (18 votes)
  • piceratops seed style avatar for user Julia Ham
    So, what is the difference between a function and a sequence? Why is a sequence discrete and a function is continuous?
    (13 votes)
    Default Khan Academy avatar avatar for user
    • duskpin sapling style avatar for user DY (darryl-yeo.com)
      In general, a function is a relations that defines an output for any input over an interval. Thus, you are often able to visualize this set of outputs on a graph as a continuous line.

      A sequence, on the other hand, is a relation that defines an output only for integer inputs. Because you cannot get an output for any value in between, you can only visualize the set of outputs on a graph as a set of discrete points.
      (27 votes)
  • aqualine ultimate style avatar for user Rishabh Choudhari
    I don't get what recursive is, can someone explain it please?
    (10 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Pete Halton
      In a recursively defined sequence, the next term is defined in terms of (excuse the pun) the terms that come before it. For example you could have a sequence where the first term is equal to 1 and where each term that follows is equal to the sum of all the previous terms: 1, 1, 2, 4, 8, 16,...
      (18 votes)
  • hopper happy style avatar for user Tushigmaa Ariunbileg
    Why is it always -1 in the end of the equation in the parenthesis? Is it possible for it to be a different number?
    (15 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user famousguy786
      We have the (k-1) multiplied to the common difference so that the formula is valid for all terms, including the first term. The first term(k=1) does not have the common ratio added to it. So for the first term (k-1) will become 0 since k=1. This is why there is (k-1) in the general formula.
      However, if the first term is divisible by the common difference, the k-1 can be changed to some other factor using algebraic manipulation. The formula remains the same but we only change or simplify the way we write it.
      For example take the sequence 2,4,6,8.....
      Its general formula is-->
      t(k)=2+2(k-1)
      However if we open the bracket we get-->
      t(k)=2+2k-2=2k
      So the (k-1) factor will always be there in the general formula but in some cases we can simplify the formula to get a different form.
      I hope that helps answer your question.
      (5 votes)
  • male robot donald style avatar for user AustinJay
    Are the Fibonacci numbers considered a sequence?
    1,1,2,3,5,8,13,21 and so on.
    (8 votes)
    Default Khan Academy avatar avatar for user
  • piceratops sapling style avatar for user Sara Rain Tree
    What is the difference between DENOTING a sequence and DEFINING a sequence? Are explicit and recursive formulas denotations or definitions?
    (9 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Débora Romagnolo
      "Define" a sequence is the act of establish a law who's govern a sequence. Like the arithmetic sequences in the video (one with the law +3 in each previous term of the sequence, and another with +4 in each previous term of the sequence). "Denoting" means showing something. Usually with a especific set of simbols and notations. Like Sal shows in the video, how do you express a sequence, using a regular notation or a function notation.
      (3 votes)
  • leafers sapling style avatar for user David
    at , why does he use k-1, why is that needed?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user moomoosnake
      k-1 is the number of differences that have to the be added to the first term to get the kth term. For example, if you want to get the 2nd term, you add 1 difference to the first term, and if you want the 3rd term, you add 2 differences to the first term (or 1 difference to the 2nd term), and so on.
      (2 votes)
  • orange juice squid orange style avatar for user Ayush Pandey
    What is the clear distinction between a sequence and a function?
    Is it the domain or any other characteristic that distinguishes?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • leaf yellow style avatar for user Howard Bradley
      I would say that a sequence is a special kind of function that has the natural numbers, ℕ, as its domain. I think you'd also have to say the domain was either infinite (defined for all natural numbers) or, if it's finite, then it's defined for the first n natural numbers. In other words you can't have gaps.
      A sequence definitely satisfies the requirement that functions be one-to-one.
      (5 votes)
  • duskpin sapling style avatar for user Nicole Bosch
    I don't understand some of the words being used.
    What does recursive, "a sub k" and explicit mean?
    Please use less complex words when explaining math problems.
    (4 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user RowanH
      a sub k seems to be the way to read aloud ak as it is written at , ie with the k below the a (ie k is the 'subscript'). I will use this 'a sub k', as I can't work out how to make subscripts.
      Explicit seems to mean either writing out the numbers directly, or providing a rule(function) for how to get any term in the sequence. For example if 'a sub k' is 2k, you can directly calculate 'a sub 5' would be 10. Something is recursive is when in order to use the rule to work out the term, you need the value of the term before. So if you used the rule
      'a sub k' = 'a sub (k-1)' + 2
      you couldn't directly calculate 'a sub 5'. You would need to know 'a sub 4'. But to calculate that, you need 'a sub 3', but for that .... etc. This is 'recursive'
      (5 votes)

Video transcript

What I want to do in this video is familiarize ourselves with the notion of a sequence. And all a sequence is is an ordered list of numbers. So for example, I could have a finite sequence-- that means I don't have an infinite number of numbers in it-- where, let's say, I start at 1 and I keep adding 3. So 1 plus 3 is 4. 4 plus 3 is 7. 7 plus 3 is 10. And let's say I only have these four terms right over here. So this one we would call a finite sequence. I could also have an infinite sequence. So an example of an infinite sequence-- let's say we start at 3, and we keep adding 4. So we go to 3, to 7, to 11, 15. And you don't always have to add the same thing. We'll explore fancier sequences. The sequences where you keep adding the same amount, we call these arithmetic sequences, which we will also explore in more detail. But to show that this is infinite, to show that we keep this pattern going on and on and on, I'll put three dots. This just means we're going to keep going on and on and on. So we could call this an infinite sequence. Now, there's a bunch of different notations that seem fancy for denoting sequences. But this is all they refer to. But I want to make us comfortable with how we can denote sequences and also how we can define them. We could say that this right over here is the sequence a sub k for k is going from 1 to 4, is equal to this right over here. So when we look at it this way, we can look at each of these as the terms in the sequence. And this right over here would be the first term. We would call that a sub 1. This right over here would be the second term. We'd call it a sub 2. I think you get the picture-- a sub 3. This right over here is a sub 4. So this just says, all of the a sub k's from k equals 1, from our first term, all the way to the fourth term. Now, I could also define it by not explicitly writing the sequence like this. I could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation. So the same exact sequence, I could define it as a sub k from k equals 1 to 4, with-- instead of explicitly writing the numbers here, I could say a sub k is equal to some function of k. So let's see what happens. When k is 1, we get 1. When k is 2, we get 4. When k is 3, we get 7. So let's see. When k is 3, we added 3 twice. Let me make it clear. So this was a plus 3. This right over here was a plus 3. This right over here is a plus 3. So whatever k is, we started at 1. And we added 3 one less than the k term times. So we could say that this is going to be equal to 1 plus k minus 1 times 3, or maybe I should write 3 times k minus 1-- same thing. And you can verify that this works. If k is equal to 1, you're going to get 1 minus 1 is 0. And so a sub 1 is going to be 1. If k is equal to 2, you're going to have 1 plus 3, which is 4. If k is equal to 3, you get 3 times 2 plus 1 is 7. So it works out. So this is one way to explicitly define our sequence with kind of this function notation. I want to make it clear-- I have essentially defined a function here. If I wanted a more traditional function notation, I could have written a of k, where k is the term that I care about. a of k is equal to 1 plus 3 times k minus 1. This is essentially a function, where an allowable input, the domain, is restricted to positive integers. Now, how would I denote this business right over here? Well, I could say that this is equal to-- and people tend to use a. But I could use the notation b sub k or anything else. But I'll do a again-- a sub k. And here, we're going from our first term-- so this is a sub 1, this is a sub 2-- all the way to infinity. Or we could define it-- if we wanted to define it explicitly as a function-- we could write this sequence as a sub k, where k starts at the first term and goes to infinity, with a sub k is equaling-- so we're starting at 3. And we are adding 4 one less time. For the second term, we added 4 once. For the third term, we add 4 twice. For the fourth term, we add 4 three times. So we're adding 4 one less than the term that we're at. So it's going to be plus 4 times k minus 1. So this is another way of defining this infinite sequence. Now, in both of these cases, I defined it as an explicit function. So this right over here is explicit. That's not an attractive color. Let me write this in. This is an explicit function. And so you might say, well, what's another way of defining these functions? Well, we can also define it, especially something like an arithmetic sequence, we can also define it recursively. And I want to be clear-- not every sequence can be defined as either an explicit function like this, or as a recursive function. But many can, including this, which is an arithmetic sequence, where we keep adding the same quantity over and over again. So how would we do that? Well, we could also-- another way of defining this first sequence, we could say a sub k, starting at k equals 1 and going to 4 with. And when you define a sequence recursively, you want to define what your first term is, with a sub 1 equaling 1. You can define every other term in terms of the term before it. And so then we could write a sub k is equal to the previous term. So this is a sub k minus 1. So a given term is equal to the previous term. Let me make it clear-- this is the previous term, plus-- in this case, we're adding 3 every time. Now, how does this make sense? Well, we're defining what a sub 1 is. And if someone says, well, what happens when k equals 2? Well, they're saying, well, it's going to be a sub 2 minus 1. So it's going to be a sub 1 plus 3. Well, we know a sub 1 is 1. So it's going to be 1 plus 3, which is 4. Well, what about a sub 3? Well, it's going to be a sub 2 plus 3. a sub 2, we just calculated as 4. You add 3. It's going to be 7. This is essentially what we mentally did when I first wrote out the sequence, when I said, hey, I'm just going to start with 1. And I'm just going to add 3 for every successive term. So how would we do this one? Well, once again, we could write this as a sub k. Starting at k, the first term, going to infinity with-- our first term, a sub 1, is going to be 3, now. And every successive term, a sub k, is going to be the previous term, a sub k minus 1, plus 4. And once again, you start at 3. And then if you want the second term, it's going to be the first term plus 4. It's going to be 3 plus 4. You get to 7. And you keep adding 4. So both of these, this right over here is a recursive definition. We started with kind of a base case. And then every term is defined in terms of the term before it or in terms of the function itself, but the function for a different term.