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

Arithmetic sequence problem

Sal finds the 100th term in the sequence 15, 9, 3, -3... Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

  • blobby green style avatar for user http://facebookid.khanacademy.org/530415458
    To the "debate on how to do this problem"
    Just use the standard form -> nth term= a1 + (n-1)*(D)
    in this case
    100th= 15 + (100-1) (-6)
    100th= -579
    where d is the common difference, a1 is the first term and n is the number of terms, then you'll never loose track of negatives!
    But Sal is just trying to get the concept across to people just being introduced to it with no intent to introduce general formulas, you should probably just take the stance that Sal is always right and is god.It has served me well. lol
    (72 votes)
    Default Khan Academy avatar avatar for user
  • female robot ada style avatar for user jabbar
    To find the sum for arithmetic sequence, sn= n(n+1)/2, it is shown (n+1)/2, can be replaced with the average of nth term and first term. How do we understand that we should not replace the "n" outside the bracket should not be replaced with nth term too.
    (6 votes)
    Default Khan Academy avatar avatar for user
  • orange juice squid orange style avatar for user Kwabzin
    Finding the 100th term (or any term that's not given) is pretty straightfoward with an explicit(ly defined) equation. But how do you do it with a recursive(ly defined) equation?

    eg with the recursive equation for this video's example: a(100)=a("subscript" 100-1) - 6

    As in, you don't have the 99th term's value so how do you find it so you can then subtract 6 from it and get the 100th term's value?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user ArDeeJ
      You want to get the analytic form (= explicitly defined) for your recursive sequence. One, kind of hand-wavey way to do it would be to calculate some amount of the first terms, try to spot the pattern and define the analytic expression.

      Another way to do it, presuming it's of the appropriate form, would be to use the first-order linear recurrence equation.

      If you have a recursively defined sequence a_n = c*a_(n-1) + d, and you're given the first term a_0, then the sequence explicitly defined is:
      a_n = a_0 * c^n + d * (c^n - 1) / (c - 1).

      Notice that if c = 1, then you have just a regular arithmetic sequence.
      (2 votes)
  • mr pants teal style avatar for user Layly Roodsari
    Why is it 15-99x6 instead of 15-100x6?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • mr pink red style avatar for user akbk209
      We're asked to seek the value of the 100th term (aka the 99th term after term # 1). We're given the first term = 15, therefore we need to find the value of the term that is 99 terms after 15. The arithmetic formula shows this by a+(n-1)d where a= the first term
      (15), n= # of terms in the series (100) and d = the common difference (-6).
      (6 votes)
  • blobby green style avatar for user kahanaram444
    What is the difference between sequence and series?...
    (2 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Hopper
      Sequence: Particular Format of Elements
      Series: Sum of the elements in a sequence.

      E.G : Sequence would be 1,2,3,4...
      E.G : Series would be 1+2+3+4...

      As you see, the Sequence helps the series. The Sequence shows the numbers, while the Series adds the numbers.

      Hope this helped! 😁
      (2 votes)
  • piceratops tree style avatar for user Nasir C
    Sal i need help on this problem −3= t/−15
    (2 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Darren
    Is there ever the -1th term in a number sequence/pattern?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • duskpin tree style avatar for user s_colin.bayless@ousd.org
    how do you find the 261 term?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Mai
    An arithmetic progression consists of 26 terms. Given the first term is 2 and the sum of the
    last 8 terms is 532. Find the 15th term of the progression. So how do I solve this?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • cacteye blue style avatar for user Jerry Nilsson
      We have an arithmetic progression
      𝐴 = {𝑎(1), 𝑎(2), 𝑎(3), ..., 𝑎(26)}

      We also have
      𝑎(19) + 𝑎(20) + 𝑎(21) + ... + 𝑎(26) = 532

      Since 𝑎(𝑛 + 1) = 𝑎(𝑛) + 𝑑, we can write this sum as
      𝑎(19) + 𝑎(19) + 𝑑 + 𝑎(19) + 2𝑑 + 𝑎(19) + 3𝑑 + ... + 𝑎(19) + 7𝑑 =
      = 8 ∙ 𝑎(19) + (1 + 2 + 3 + ... + 7)𝑑 =
      = 8 ∙ 𝑎(19) + 28𝑑 = 532 ⇒
      ⇒ 𝑎(19) = (532 − 28𝑑)∕8 = 66.5 − 3.5𝑑

      𝑎(𝑛) = 𝑎(1) + (𝑛 − 1)𝑑 ⇒ 𝑎(19) = 𝑎(1) + 18𝑑
      𝑎(1) = 2 ⇒ 𝑎(19) = 2 + 18𝑑

      Thereby, 66.5 − 3.5𝑑 = 2 + 18𝑑 ⇒
      ⇒ 𝑑 = (66.5 − 2)∕(18 + 3.5) = 3

      𝑎(15) = 𝑎(1) + 14𝑑 = 2 + 14 ∙ 3 = 44
      (2 votes)
  • blobby green style avatar for user nickqt
    I tend to do things differently. When watching these videos, I attempt to do them before watching. Here's what I did, although it is slightly different, it gives the same answers.

    Term 0 | Value 21
    Term 1 | Value 15
    Term 2 | Value 9
    ...

    Since Term 0 is Value 21, and Term 1 is 15, each term can be multiplied by -6, with 21 being added to that product.

    Term N | (N * -6)+21

    Term 100 | (100 * -6) + 21
    Term 100 | -600 + 21 = -579

    Same answer, different method. For whatever it's worth.

    (3 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- [Instructor] We are asked what is the value of the 100th term in this sequence, and the first term is 15, then nine, then three, then negative three. So let's write it like this in a table. So if we have the term, just so we have things straight, and then we have the value, and then we have the value of the term. I'll do a nice little table here. So our first term we saw is 15. Our second term is nine. Our third term is three. I'm just really copying this down, but I'm making sure we associate it with the right term. And then our fourth term, our fourth term is negative three. And they wanna ask, they want us to figure out what the 100th term of this sequence is going to be. So let's see what's happening here, if we can discern some type of pattern. So we went from the first term to the second term, what happened? 15 to nine, looks like we went down by six. It's always good to think about just how much the numbers changed by. That's always the simplest type of pattern. So we went down by six, we subtracted six. Then to go from nine to three, well we subtracted six again. We subtracted six again. And then to go from three to negative three, well we, we subtracted six again. We subtracted six again. So it looks like every term, you subtract six. So the second term is going to be six less than the first term. The third term is going to be 12 minus from the first term, or six subtracted twice. So in the third term, you subtract a six twice. In the fourth term, you subtract six three times. So whatever term you're looking at, you subtract six one less than that many times. Let me write this down just so, notice when your first term, you have 15 and you don't subtract six at all, or you could say you subtract six zero times. So you could say this is 15 minus six times or let me write it better this way, minus zero times six. That's what that first term is right there. What's the second term? This is 15, it's just we just subtracted six once, or you could say minus one times six. Or you could say plus one times negative six, either way, we're subtracting the six once. Now what's happening here? This is 15, this is 15 minus two times negative six, or sorry minus two times six, minus two times six. We're subtracting a six twice. What's the fourth term? This is 15 minus, we're subtracting the six three times from the 15, so minus three times six. So if you see the pattern here, when our term, when we have our fourth term, we have the term minus one right there, the fourth term we have a three, the third term we have a two, the second term we have a one. So if we had the nth term, if we just had the nth term here, what's this going to be? It's going to be 15 minus, you see it's going to be n minus one right here, right when n is four, n minus one is three. When n is three, n minus one is two. When n is two, n minus one is one. When n is one, n minus one is zero. So we're going to have, this term right here is n minus one, so minus n minus one times six. So if you wanna figure out the 100th term of this sequence, I didn't even have to write it in this general term, you could just look at this pattern. It's going to be, and I'll do it in pink, the 100th term in our sequence, I'll continue our table down, is gonna be what? It's going to be 15 minus 100 minus one, which is 99, times six, right? I just followed the pattern. One, you had a zero here, two, you had a one here, three, you had a two here, 100, you're gonna have a 99 here. So let's just calculate what this is. What's 99 times six? So 99 times six, actually you could do this in your head. You could say that's going to be six less than 100 times six, which is 600, and six less is 594. But if you didn't wanna do it that way, you just do it the old-fashioned way. Six times nine is 54, carry the five. Nine times six or six times nine is 54. 54 plus five is 594. So this right here is 594, and then to figure out what 15, so we wanna figure out, we wanna figure out what 15 minus 594 is, and this can sometimes be confusing, but the way I always process this in my head is I say that this is the exact same thing as the negative of 594 minus 15. And if you don't believe me, distribute out this negative sign. Negative one times 594 is negative 594, negative one times negative 15 is positive 15. So these two statements are equivalent. This is much easier for my brain to understand. So what's 594 minus 15? You should do it, we could do this in our head. 594 minus 14 would be 580, and then 580 minus one more would be 579. So that right there is 579, and then we have this negative sign sitting out there. So our the 100th term in our sequence will be negative 579.