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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, plus, 3, plus, 5, plus, 7, plus, 9.

  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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:
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:
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, plus, 5, plus, 7, plus, point, point, point, plus, 401.

Find the values of a, start subscript, 1, end subscript and a, start subscript, n, end subscript for this series.
a, start subscript, 1, end subscript, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
a, start subscript, n, end subscript, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Find the value of n for this series.
n, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Find the sum of 3, plus, 5, plus, 7, plus, point, point, point, plus, 401
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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

Try it yourself

Problem 1
Find the sum.
11, plus, 20, plus, 29, plus, point, point, point, plus, 4052, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Nice! Try another one!
Problem 2
Find the sum.
10, plus, left parenthesis, minus, 1, right parenthesis, plus, left parenthesis, minus, 12, right parenthesis, plus, point, point, point, plus, left parenthesis, minus, 10, comma, 979, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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.
    (34 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
      (7 votes)
  • blobby green style avatar for user Meghna Prakash
    How do we calculate the value of n
    (5 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
      (22 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
  • leaf green style avatar for user 廣涵 王
    What's exactly the difference between "progression", "sequence" and "series"?
    (6 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.
    (7 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)
  • blobby green style avatar for user lovourpryor
    How is it 1000 for the last one?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • 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)
  • 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)
    Default Khan Academy avatar avatar for user
    • 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.
      (4 votes)
  • 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)
    Default Khan Academy avatar avatar for user
    • 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)
  • mr pants teal style avatar for user Shraavani S Shanbhag
    I did not understand this sum:
    Find the sum; 11+20+29+...+4052.

    Please let me know,how to solve this?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user PeterRI
      In order to solve this series we have to make pairs.

      1. Find the number of total numbers in the equation.
      So because we start at 11 we subtract 11 from 4052. Then we get 4041, Next we divide by 9 because that is the increments that this is increasing by. This gives us 449 but we must remember that at the beginning we took out the 11, so we have 450 total numbers. (Wow what are the chances of getting such a nice number😮)

      2. Next we add the smallest and biggest numbers together. We do this 225 times. The smallest and biggest numbers are 11 and 4041. when we add them together we get 4063. We get this number 225 times because after 11 and 4052 are used we go to the next numbers 20 and 4043. You will see that we will get the same number every time because the small number increases by 9 and the big number decreases by 9.

      3. Now we just need to do some simple arithmetic. 4063*225= 914175

      4. Check the [Explain] button before you ask a question because it says what I am saying here but with equations instead of words.
      (4 votes)