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Arithmetic series

Walk through a guided practice where you'll start by finding a simple sum and end by evaluating finite arithmetic series.
Let's start with an addition problem.

Find the sum of 1+3+5+7+9.

  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Awesome! You just found the sum of a small arithmetic series. It only had 5 terms. But, what if it had one million terms? We'd want a formula for sure. Thankfully, we've already learned of such a formula.
Identify the formula for the sum of an arithmetic series.
Choose 1 answer:

Sweet! So you remember the formula. Now let's make sure we remember how to apply it.
Choose the answer that shows the formula correctly used to find the sum you found.
Choose 1 answer:

Alright, so we're feeling good so far. Let's try to use the formula to find the sum of an arithmetic series that would be tedious to calculate by hand.

Consider the series 3+5+7++401.

Find the values of a1 and an for this series.
a1=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
an=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Find the value of n for this series.
n=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Find the sum of 3+5+7++401
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Wow! Okay, looks like you've got this.

Try it yourself

Problem 1
Find the sum.
11+20+29++4052=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Nice! Try another one!
Problem 2
Find the sum.
10+(1)+(12)++(10,979)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to join the conversation?

  • aqualine seed style avatar for user Cathryn  Newsom
    I do not understand how to find the n value. This did not make any sense to me. I need a formula or an explanation on how to find the n.
    (44 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Michael Johnson
      Another way you can do it is to come up with a function, and plug in your final value and solve for x. For example, when finding the sum of 3 + 5 + 7 + .... + 401 It might help to start out with a little chart:

      x f(x)
      1 3
      2 5
      3 7
      4 9

      And from that you might intuit
      f(x) = 2x + 1

      Then plug in your final term, which is 401, and solve for x
      401 = 2x + 1
      400 = 2x
      x = 200 tada

      Sometimes you’ll see a slightly different, but equivalent function. For example, the same 3 + 5 + 7 + .... + 401 you might first identify as
      f(x) = 3 + 2(x-1)
      and that’s fine, because mathematically it’s the same function. Even if you don’t simplify it to 2x + 1 you still find x = 200 when you plug in your final term in the sequence.
      401 = 3 + 2(x-1)
      401-3 = 2(x-1)
      398 = 2 (x-1)
      398/2 = x-1
      199 = x - 1
      x = 200
      (21 votes)
  • blobby green style avatar for user Meghna Prakash
    How do we calculate the value of n
    (6 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Guru Adi
      Take the LAST number in the sequence MINUS the FIRST number in the sequence.
      DIVIDE that value by the pattern in the sequence.
      How do you find the pattern? Ask, "How do I get from the first term, to the second term?"
      Hope that helps
      (35 votes)
  • blobby green style avatar for user Bill Lagarde
    How do you determine the value of n? I don't think this was every explained.
    (9 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user elisabeth.rose.siena
    For problem 2: how did you find out the n was 450? I tried doing An-A1/2 and adding one for the first term and I got a different number.
    (9 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Manasa
      it's like this...
      let's say we have an arithmetic sequence that goes like 2, 4, 6, 8, ..., 262
      lets take first term as a=2
      common difference as d=2
      last term or nth term as a(n)=262
      we know the formula for nth term is a(n)=a+(n-1)d
      so here, 262=2+(n-1)2
      so n-1 = (262-2)/2 = 260/2 = 130
      so n= 130+1 =131
      Sal explained it in a non-formula, less mathematical more logic-based kind of way, but this is the mathematical basis for it
      hope it helps :)
      (7 votes)
  • leaf green style avatar for user 廣涵 王
    What's exactly the difference between "progression", "sequence" and "series"?
    (6 votes)
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  • aqualine ultimate style avatar for user Ash
    Why is this so confusing?

    HELP
    (4 votes)
    Default Khan Academy avatar avatar for user
    • old spice man green style avatar for user Enwesi Chiamaka
      Okay, for each series of terms given above, you fish out the 'first term(a), the common difference(d)->which is gotten by subtracting the previous term from the next {like now, 1+2+3+4...+10> in this, to get the common difference, you would subtract 1 from 2 or subtract 2 from 3. I'm sure you get the point, the subtraction should be progressive. This tells us that our common difference in this case is .....drumrolls....1, yes you got it!} and the number of terms (n) which is gotten by looking at the last term given and substituting it in this formula of "nth" terms, Tn= a+(n-1)d={the last term you see there}. By doing so, you would find your "n" successfully. With that formula, note that 'a', 'd' and 'n' are involved! When you are done doing all of this, you will then use the final formula for the Sum of the terms> Sn=n/2{2a+(n-1)d} to find the sum of the whole terms given in the series provided. I am confident that you get it now! If there is any other problem, please do not hesitate to reply or reach out!
      (7 votes)
  • blobby green style avatar for user lovourpryor
    How is it 1000 for the last one?
    (6 votes)
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  • purple pi purple style avatar for user Tanya Nacar
    on problem 1, could someone tell me how they found out the number of terms was 450?
    (2 votes)
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    • duskpin sapling style avatar for user Vu
      11+20+29+...+4052

      They found n (the last term) by set 4052 into the explicit formula. So to find n you must know how to formulate the formula from the sequence.

      This has the initial of 11 and common difference of 9, so a(n)=11+9(n-1). So to find what n is when a(n) = 4052, you set 4052=11+9(n-1) and solve for n. They didn't explain but that's how you would find n in that problem.
      (5 votes)
  • duskpin ultimate style avatar for user melvinakhalaf
    can somebody pleeeease explain how to find n like in question 4b I read the comments and the explanation but couldn't conclude a formula
    (4 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Manasa
      it's like this...
      let's say we have an arithmetic sequence that goes like 2, 4, 6, 8, ..., 262
      lets take first term as a=2
      common difference as d=2
      last term or nth term as a(n)=262
      we know the formula for nth term is a(n)=a+(n-1)d
      so here, 262=2+(n-1)2
      so n-1 = (262-2)/2 = 260/2 = 130
      so n= 130+1 =131
      Sal explained it in a non-formula, less mathematical more logic-based kind of way, but this is the mathematical basis for it
      hope it helps :)
      (1 vote)
  • female robot grace style avatar for user Bennett Powell
    What the heck does this mean: Find the sum of first 335 terms

    A(sub)1 = 2
    A(sub)i = A(sub< i -1 >) -3

    What does the "i" mean?
    (2 votes)
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    • starky ultimate style avatar for user KLaudano
      The subscripted numbers denote the number of the term in series A. The subscripted i can be any number other than 1. (The first term is separately defined.) The subscripted i-1 refers to the term immediately before term i.

      Based on the example you gave, here are the first few terms. (The underscore is used to show a subscript.)
      A_1 = 2
      A_2 = A_1 - 3 = 2 - 3 = -1
      A_3 = A_2 - 3 = -1 - 3 = -4
      A_4 = A_3 - 3 = -4 - 3 = -7
      (4 votes)